SAT Advanced Math practice questions
Quadratics, exponentials, polynomials and nonlinear systems: the questions that separate strong Math scores from great ones.
1571 free questions, each with a full worked solution and an explanation of what each wrong answer means.
Practise this domain adaptively →Easy (314)
- Evaluating linear functions
If f(x) = 3x − 7, what is the value of f(5)? - Difference of squares
What are the solutions to x² − 9 = 0? - Factoring quadratic expressions
Which of the following expressions is equivalent to x² + 7x + 12 ? - Evaluating a polynomial function at a negative input
The function g is defined by g(x) = x³ + 2x. What is the value of g(−2) ? - Rational exponents and radical form
For x > 0, which of the following is equivalent to ∛(x⁵) ? - Zeros of a quadratic given in factored form
The function h is defined by h(x) = (x − 6)(x + 2). What is one value of x for which h(x) = 0 ? - Expanding a product of binomials
Which expression is equivalent to (x + 3)(x + 5)? - Factoring a difference of squares
Which expression is equivalent to x² − 9? - Reading the vertex from vertex form
The function f is defined by f(x) = (x − 3)² + 5. What are the coordinates of the vertex of the graph of f in the xy-pla… - Distributing a constant
Which expression is equivalent to 3(2x − 5)? - Combining like terms
Which expression is equivalent to (4x + 7) + (2x − 3)? - Evaluating a quadratic model
What is the height of the ball, in metres, 3 seconds after it is thrown? - Recognising exponential decay
Which of the following functions represents exponential decay? - Monomial times a trinomial
Which expression is equivalent to 2x(x² − 3x + 4)? - Evaluating a quadratic at a negative input
The function f is defined by f(x) = −x² + 4x. What is the value of f(−1)? - Solving a cube equation
If x³ = 27, what is the value of x? - Distributing a negative and combining
Which expression is equivalent to −2(3x − 4) + 5x? - Evaluating an exponential function
The function f is defined by f(x) = 3 · 2ˣ. What is the value of f(4)? - The negative square root
If x² = 100 and x < 0, what is the value of x? - Adding expressions with three variables
Which expression is equivalent to (3a + 2b − c) + (a − 5b + 4c)? - Distributing and subtracting
Which expression is equivalent to 5x − 2(x − 3)? - Adding two function values
The function f is defined by f(x) = x² − 3x. What is the value of f(0) + f(3)? - Positive root of a factored equation
If (x − 3)(x + 8) = 0, what is the positive value of x that satisfies the equation? - Exponential function from a table
Which equation defines the function f shown in the table? - Factoring out x
The equation x² − 5x = 0 has two solutions. What is the positive solution? - Evaluating a quadratic model
The height, in metres, of a drone t seconds after it is launched is modelled by h(t) = 4t − t². What is the drone's heig… - Isolating x² by subtracting
If x² + 1 = 26, what is the positive value of x? - Positive number from its square
The square of a positive number is 64. What is the number? - Dividing monomials
For x ≠ 0, which expression is equivalent to 6x²/(2x)? - Evaluating a square-root function
The function g is defined by g(x) = √(x + 5). What is the value of g(4)? - Product of conjugate binomials
Which expression is equivalent to (x − 7)(x + 7)? - Evaluating x² − x at a negative input
The function p is defined by p(x) = x² − x. What is the value of p(−2)? - Evaluating a scaled square
The function q is defined by q(x) = 3x². What is the value of q(2)? - Undoing a square root
If √x = 9, what is the value of x? - Cubing a negative input
The function r is defined by r(x) = x³. What is the value of r(−2)? - Distribute, then combine
Which expression is equivalent to 5(2x − 3) − 4x? - Evaluating a shifted exponential
The function s is defined by s(x) = 2ˣ + 3. What is the value of s(2)? - Clearing a denominator under x²
If x²/4 = 9 and x > 0, what is the value of x? - Side of a tile from a combined area
Three identical square tiles have a combined area of 48 square inches. What is the side length, in inches, of one tile? - Combining like terms
Which expression is equal to 3x + 5x for all values of x? - Evaluating a small power
What is the value of 2³? - Evaluating a quadratic expression
If x = 4, what is the value of x² − 1? - Distributing a number over a sum
Which expression is equal to 2(x + 3) for all values of x? - Square root of a perfect square
What is the value of √49? - Positive square root from a squared equation
If x² = 36 and x > 0, what is the value of x? - Positive solution of a simple quadratic
What is the positive solution of the equation x² − 9 = 0? - Doubling count as an exponential equation
A single cell splits into 2 cells every hour, so after h hours there are 2ʰ cells. After how many hours are there 16 cel… - Side of a square from its area
A square has an area of 81 square centimetres. What is the length, in centimetres, of one side of the square? - Nonzero solution of a factored equation
If x(x − 3) = 0 and x ≠ 0, what is the value of x? - Evaluating a squaring function
If f(x) = x², what is the value of f(3)? - Counting x-intercepts of a parabola above the axis
In the xy-plane, at how many points does the graph of y = x² + 4 cross the x-axis? - Reading the vertex from vertex form
In the xy-plane, what is the vertex of the graph of y = (x − 2)² + 1? - Direction a parabola opens
Which of the following best describes the graph of y = −x² in the xy-plane? - Substituting into a linear expression
When x = 3, what is the value of 2x + 4? - Subtracting like terms
Which expression is equal to 4x − x for all values of x? - Square of a negative number
What is the value of (−2)²? - Evaluating a quadratic expression with two terms
If y = 5, what is the value of y² + y? - Writing a repeated product as a power
Which expression is equal to x · x · x? - Counting the solutions of x squared equals zero
How many different real numbers x satisfy the equation x² = 0? - Quadratic with a coefficient and no linear term
If 5x² = 20 and x > 0, what is the value of x? - Positive root of a factored quadratic
For what positive value of x is (x − 4)(x + 2) equal to 0? - Number from its square root
The square root of a positive number is 4. What is the number? - A number times itself
A positive number multiplied by itself gives 49. What is the number? - Behaviour of a growing exponential
As x increases, what happens to the value of y = 2ˣ? - Reading an exponential rule at a small input
Using the equation y = 3ˣ, what is the value of y when x = 2? - Positive input for a given output of a squaring function
The function f is defined by f(x) = x². What positive value of x gives f(x) = 16? - Naming the shape of a quadratic graph
In the xy-plane, the graph of y = x² is a curve. What is this curve called? - Writing a multiple in terms of a variable
A box holds n pens. Which expression gives the number of pens in 3 such boxes? - Sum of two squares
What is the value of 3 squared plus 4 squared? - Repeated doubling by folding
A sheet of paper is folded in half, then in half again, and so on, four times in all. How many layers thick is the folde… - Doubling an expression with a known value
If 3x = 12, what is the value of 6x? - Dividing a square by its base
If x = 10, what is the value of x² ÷ x? - Finding an exponent
3 raised to what power equals 81? - Edge of a cube from its volume
A cube has a volume of 27 cubic centimetres. What is the length, in centimetres, of one edge of the cube? - Counting real solutions of a negative square
How many real numbers x satisfy x² = −1? - Solution of a squared binomial equal to zero
If (x − 1)² = 0, what is the value of x? - Larger solution of a quadratic with a zero root
The equation x² = 5x has two solutions. What is the larger of the two? - Starting value of a quadratic height model
A ball's height, in metres, t seconds after it is dropped is given by h(t) = 20 − 5t². What is the ball's height at t = … - Second x-intercept of a symmetric parabola
In the xy-plane, the graph of y = x² − 4 crosses the x-axis at x = 2 and at one other value of x. What is that other val… - Evaluating a quadratic with a coefficient
The function g is defined by g(x) = 3x². What is the value of g(2)? - Vertical shift of a parabola
In the xy-plane, how does the graph of y = x² + 1 compare with the graph of y = x²? - Area of a rectangle as an expression
A rectangle has a length of 2x units and a width of 3 units. Which expression gives its area, in square units? - Evaluating a perimeter formula
The perimeter of a square with side length s is given by 4s. What is the perimeter of a square with side length 7? - Writing a repeated numeric product as a power
Which of the following is the same as 5 × 5 × 5? - Square of a decimal
What is the value of 0.5²? - Half of a half as a fraction
A cake is cut in half, and one of the halves is cut in half again. Which fraction of the whole cake is one of the smalle… - All solutions of a simple quadratic
Which of the following lists all the real solutions of x² = 49? - Which exponent gives 1
A student raises 2 to a whole-number power and gets 1. Which power did the student use? - Cube that is negative
A number multiplied by itself three times gives −8. Which number is it? - Number whose square root is zero
Which number has a square root equal to 0? - Negative exponent that gives a reciprocal
Which power of 3 is equal to 1/3? - Mirror-image points on a parabola
The graph of y = x² passes through the point (3, 9). Because the graph is symmetric about the y-axis, which other point … - Point on the graph of a squaring function
Which of the following points lies on the graph of y = x² in the xy-plane? - Lowest point of a lifted parabola
In the xy-plane, the graph of y = x² + 5 has a lowest point. What is the y-coordinate of that point? - Difference between two values of an exponential
For the function y = 2ˣ, how much larger is the value of y at x = 3 than at x = 2? - Perimeter of a square as an expression
A square has side length 3x. Which expression gives its perimeter? - Total from three related amounts
Kai has n marbles. Ali has 3 times as many marbles as Kai, and Bea has 2 more marbles than Kai. Which expression represe… - Constant term after expanding
Expanding 4(x + 2) gives 4x plus a constant. What is that constant? - Adding like terms
A student adds 2a + 2a + 2a. Which expression is the sum? - Coefficient of pi in a circle area
The area of a circle with radius r is πr². When r = 3, the area can be written as kπ. What is the value of k? - Equation with exactly two real solutions
Which of the following equations has exactly two real solutions? - Positive number times itself
The product of a positive number and itself is 49. What is the number? - Recovering a number from its square plus one
Fen squares a positive number and then adds 1, getting 10. What number did Fen start with? - Power of ten
The number 1,000 is 10 raised to what power? - Solution of a parabola-and-horizontal-line system
Which of the following pairs (x, y) is a solution of the system y = x² and y = 4? - Describing a doubling table
The table shows values of a function g. x | g(x) 0 | 1 1 | 2 2 | 4 3 | 8 Which statement describes how g(x) changes ea… - Distance fallen from a quadratic model
An object dropped from a tower falls 5t² metres in the first t seconds. How many metres does it fall in the first 2 seco… - Cube volume function
A cube with edge length e has volume V(e) = e³. What is the value of V(4)? - Parabola that opens downward
In the xy-plane, the graph of y = x² opens upward. Which of the following equations has a graph that opens downward? - Coefficient after combining terms
The expression 6x − 2x + x simplifies to kx, where k is a constant. What is the value of k? - Pencils in three boxes
A box holds 2y pencils. Which expression gives the number of pencils in 3 such boxes? - Single exponent from a product of powers
When 2³ × 2² is written as a single power of 2, what is the exponent? - Length of a rectangle from its area
The area of a rectangle is x² + 3x, and its width is x. Which expression gives its length? - Constant in a sum of binomials
The sum (x + 4) + (x − 1) simplifies to 2x + c, where c is a constant. What is the value of c? - Positive solution of a simple square equation
The equation x² = 100 has two solutions. What is the positive one? - Sum of the solutions of a factored equation
The equation (x − 2)(x − 3) = 0 is true for exactly two values of x. What is the sum of those two values? - Exponent that gives 27
The equation 3ˣ = 27 is true for exactly one value of x. What is that value? - Why a square cannot be negative
A student claims that the equation x² = −4 has a real solution. Which statement explains why the student is wrong? - Square with the area of a rectangle
A square has the same area as a rectangle that is 2 centimetres by 8 centimetres. What is the side length of the square,… - Matching a function to a table
The table shows three values of a function f. x | f(x) 1 | 1 2 | 4 3 | 9 Which of the following could define f? - Height of a thrown ball after one second
The height h, in metres, of a ball t seconds after it is thrown upward is h(t) = 20t − 5t². What is the height of the ba… - Sign of a squaring function
The function g is defined by g(x) = x². Which of the following is true for every real number x? - Balance that triples each year
A savings balance starts at $2 and triples every year. What will the balance be, in dollars, after 3 years? - Starting value of an exponential model
The number of visitors to a website t days after launch is modelled by 3·2ᵗ. How many visitors were there on the day of … - Choosing the graph shape of a quadratic
In the xy-plane, the graph of y = x² + 2 has which of the following shapes? - Two inputs with the same output
For the function f(x) = x², which pair of inputs gives the same output? - Evaluating a falling-object model
The height h, in metres, of a stone t seconds after it is dropped from a 100-metre cliff is h(t) = 100 − 10t². What is h… - Recognising exponential growth in words
A quantity starts at 5 and is multiplied by 3 every hour. Which type of function models the quantity over time? - Coefficient of a product of monomials
The product 3x · 4x is equal to kx², where k is a constant. What is the value of k? - Perimeter of an equilateral triangle
An equilateral triangle has sides of length x + 2. Which expression gives its perimeter? - Coefficient after distributing x
Expanding x(x + 5) gives x² + kx. What is the value of k? - Constant inside a factored expression
The expression 6x + 9 can be written as 3(2x + c), where c is a constant. What is the value of c? - Volume of a cube with edge x squared
A cube has edge length x². Which expression gives the volume of the cube? - Positive number whose square minus nine is zero
A positive number is squared, and then 9 is subtracted, giving 0. What is the number? - Only real cube root of eight
The equation x³ = 8 has exactly one real solution. What is it? - Square whose perimeter equals its area
A square with side length s has perimeter 4s and area s². For what positive value of s are the perimeter and the area nu… - Counting real solutions of x squared plus one
How many real solutions does the equation x² + 1 = 0 have? - Sum of x-coordinates of symmetric solutions
The system y = x² and y = 9 has two solutions. What is the sum of the x-coordinates of the two solutions? - Increase in area from a change in side
The area A of a square with side length s is given by A(s) = s². By how much does the area increase when s changes from … - Area of a pen from a quadratic model
A rectangular pen is built with 20 metres of fencing. If its width is w metres, its area A, in square metres, is given b… - Height after two halving bounces
A ball bounces to half its previous height each time it hits the ground. It is dropped from a height of 16 metres. How h… - Shifting a parabola upward
The graph of y = x² is shifted upward by 3 units. Which equation describes the new graph? - Subtracting a binomial
A rope of length 5t is cut, and a piece of length 2t + 1 is removed. Which expression gives the length of the remaining … - Sharing an expression equally
A total of 8y sweets is shared equally among 4 children. Each child receives ky sweets, where k is a constant. What is t… - Combining terms with a constant
Which expression is the simplified form of 2m + 3m + 4? - Evaluating with the correct order of operations
What is the value of 2n² when n = 3? - Checking a proposed solution
A student claims that x = 5 is a solution of x² − 25 = 0. Which statement is correct? - Side of a square from a doubled area
Two identical square tiles together cover 50 square centimetres. What is the side length of one tile, in centimetres? - Number from square minus itself
A positive number squared, minus the number itself, equals 12. What is the number? - Solutions of a product equal to zero
Which of the following are all the solutions of x(x − 7) = 0? - Square root of a fraction
If x² = 1/4 and x is positive, what fraction is x? - Base of a square number
36 is the square of which positive whole number? - Reading an exponential model at a given time
The number of leaves on a young plant w weeks after planting is modelled by L(w) = 4·3ʷ. How many leaves does the model … - Comparing doubling with squaring
For x = 5, which of the following correctly compares 2x and x²? - Starting value from a linear-looking model of temperature
A cup of tea cools so that its temperature T, in °C, after m minutes is modelled by T(m) = 20 + 60(1/2)^(m/10). What was… - Identifying a quadratic from a graph description
The graph of a function in the xy-plane is a U-shaped curve with its lowest point at the origin. Which of the following … - Total cost of two kinds of item
Shirts cost s dollars each and hats cost h dollars each. Which expression gives the total cost of 3 shirts and 2 hats? - Loaves left after selling
A shop starts the day with 10 loaves and sells 2 loaves every hour. Which expression gives the number of loaves left aft… - Spotting the non-equivalent expression
Which of the following is NOT equivalent to 6x? - Constant term of a squared binomial
When (x + 3)(x + 3) is expanded, what is its constant term? - Coefficient after a multiply and a subtract
A number n is multiplied by 5, and then 2n is subtracted from the product. The result is equal to kn, where k is a const… - Number whose square is four times itself
The square of a positive number is 4 times the number. What is the number? - Positive number from twice plus three equals square
Twice a positive number, plus 3, is equal to the square of the number. What is the number? - Product of a number and its successor
The product of a positive number and the number 1 greater than it is 12. What is the number? - Both numbers whose square is 49
The equation x² = 49 is solved by which pair of numbers? - Positive number from a halved square
A positive number is squared, and the result is halved to give 18. What is the number? - Negative input for a given output
For the function h(x) = x², which negative input gives the output 16? - Greatest value of a downward parabola
The function p is defined by p(x) = −x². What is the greatest value that p(x) can take? - Car value after two halvings
A car's value halves every 4 years. It is worth $8,000 now. What will it be worth, in dollars, 8 years from now? - Extra tiles between two square patterns
The number of tiles in the nth pattern of a sequence is n². How many more tiles are in the 5th pattern than in the 4th? - Coefficient of a after combining
The expression 4a + 3b − a simplifies to ka + 3b, where k is a constant. What is the value of k? - Perimeter of a rectangle in terms of its sides
A rectangle has length L and width W. Which expression gives its perimeter? - Total pencils from packs and loose pencils
Ben buys 2 packs of pencils, each holding x pencils, and gets 3 loose pencils with each pack. He then buys one more pack… - Area of a square with side 2x
The side length of a square is 2x. Which expression gives the area of the square? - Coefficient of x after expanding
When 5(x − 1) is expanded, what is the coefficient of x? - Expression for three fewer than twice an amount
Tia has t stickers. Raj has 3 fewer stickers than twice as many as Tia. Which expression gives the number of stickers Ra… - Equation solved by x = 3
Which of the following equations has x = 3 as a solution? - Constant giving exactly one solution
The equation x² = k, where k is a constant, has exactly one real solution. What is the value of k? - Product of two squares
If x² = 2 and y² = 8, what is the value of x²y²? - Sum of the solutions of a square equal to a double
The equation x² = 2x has two solutions. What is their sum? - Possible y on a circle for a given x
If x² + y² = 25 and x = 3, what is one possible value of y? - Two equal positive factors
The product of two equal positive numbers is 36. What is each of the numbers? - Comparing outputs at opposite inputs
The function f is defined by f(x) = x² + 1. Which of the following is true? - Expression for a doubling population
A population of 100 bacteria doubles every hour. Which expression gives the population after h hours? - Value of an exponential function at zero
For the function f(x) = 5ˣ, which of the following is the value of f(0)? - Water left after a quadratic drain
The amount of water, in litres, left in a bucket t minutes after a plug is pulled is modelled by w(t) = 12 − t². How man… - Choosing the graph that keeps rising faster
Which of the following functions has a graph in the xy-plane that rises more and more steeply as x increases? - Evaluating a quadratic at a negative input
The function q is defined by q(x) = x² − 5. What is the value of q(−3)? - Recognising a table that is not linear
The table shows values of a function. x | y 0 | 1 1 | 3 2 | 9 3 | 27 Which statement about the function is true? - Legs on cats and birds
A farm has c cats and b birds. Which expression gives the total number of legs on all the cats and birds? - Books left on a shelf
A shelf holds 9y books, and 4y of them are removed. Which expression gives the number of books left? - Perimeter with an expression for the length
A rectangle has length 3x + 1 and width 2. Which expression gives its perimeter? - Repeated multiplication as a power
When x · x · x · x is written as a single power xⁿ, what is the value of n? - Perimeter plus area of a square
A square has side length a. When a = 3, what is the value of its perimeter plus its area? - Fencing round a square field
A square field has an area of 64 square metres. How many metres of fencing are needed to go all the way round it? - Positive number with a given cube
The cube of a positive number is 125. What is the number? - Positive root of a factored equation
The equation (x − 1)(x + 4) = 0 has two solutions. What is the positive one? - Positive number that squares to a quarter
A positive number squared is 0.25. What is the number? - Positive value where a cube equals four squares
For what positive value of x is x³ equal to 4x²? - Distance fallen in the first second
The distance d, in metres, that an object falls in t seconds is given by d(t) = 4.9t². How many metres does it fall in t… - Next value in a doubling table
The table shows values of a function. x | y 0 | 5 1 | 10 2 | 20 If the pattern continues, what is the value of y when … - Area from a width-based function
The function A(w) = w(w + 2) gives the area of a rectangle whose length is 2 more than its width w. What is the area whe… - Graph that never meets the x-axis
Which of the following equations has a graph in the xy-plane that never crosses or touches the x-axis? - Constant after subtracting a bracket
The expression 7 − (x − 2) simplifies to c − x, where c is a constant. What is the value of c? - Pencils kept after giving some away
Ola buys 3 packs of pencils, each holding x pencils, and then gives away 5 pencils. Which expression gives the number of… - Coefficient after doubling a square term
The expression 4y² · 2 is equal to ky², where k is a constant. What is the value of k? - Half of an expression
Which of the following is equal to half of 6x? - Eggs for cakes and glaze
A recipe uses a eggs for each cake and b eggs for the glaze that covers all the cakes. If a = 2 and b = 5, how many eggs… - Checking which value solves a quadratic
Which of the following values of x satisfies x² = 2x + 8? - Gap between the two roots of a square equation
The equation x² = 49 has two solutions. By how much do they differ? - Cube root of 64
What is the cube root of 64? - Equation with two opposite solutions
Which of the following equations has both 3 and −3 as solutions? - Next power of ten
If 10ˣ = 100, what is the value of 10^(x + 1)? - Difference between two square rugs
A square rug's area, in square metres, is the square of its side length in metres. How many square metres larger is a ru… - Function that doubles with each unit step
For which of the following functions does the value double every time x increases by 1? - Where a thrown stone lands
A stone thrown across a river is at a height of 100 − x² metres when it is x metres horizontally from the thrower. How m… - Input that minimises a squared difference
For which value of x is the value of y = (x − 5)² smallest? - Starting value of a quadratic model
The number of visitors to a stall t hours after it opens is modelled by v(t) = 3t² + 7. How many visitors does the model… - Choosing the growing exponential
Which of the following functions increases as x increases? - Coefficient after combining like terms
When 7y − 2y + y is simplified, what is the coefficient of y? - Positive solution of a factored equation
What is the positive solution of the equation x(x − 6) = 0? - Ratio of two values of an exponential
The value of 2ˣ when x = 5 is how many times the value of 2ˣ when x = 3? - Value of a simplified expression
What is the value of 4y + 3y when y = 2? - Positive number from a squared-and-doubled clue
A positive number is squared and the result is doubled, giving 50. What is the number? - Value of an exponential expression
The number of leaves on a young plant is modelled by L(w) = 3 · 2ʷ, where w is the number of weeks since it sprouted. Ac… - Value of a monomial
What is the value of 5a² when a = 3? - Value after collecting terms
What is the value of 2b + 3b − b when b = 4? - Side of a square patio
A square patio has an area of 64 square feet. What is the length of one side of the patio, in feet? - Positive root of x squared minus 100
What is the positive value of x for which x² − 100 = 0? - Growth after one step of an exponential
A population is modelled by P(t) = 4 · 3ᵗ. By how much does P increase from t = 0 to t = 1? - Recognising a quadratic from its rule
A function assigns to each number x the value x² + 1. Which of the following describes the graph of this function in the… - Value of a cubic monomial
What is the value of 3m³ when m = 2? - Difference of like terms evaluated
What is the value of 9k − 4k when k = 5? - Square minus a constant equal to nine
For what positive value of x is x² − 7 equal to 9? - Side of a square tile from its area
A square tile has an area of 36 square inches. How many inches long is each side of the tile? - Starting value of an exponential model
The number of subscribers to a newsletter is modelled by S(m) = 6 · 2ᵐ, where m is the number of months since it began. … - Height of a ball after two bounces
A ball is dropped from a height of 100 centimetres. After each bounce it rises to 80% of the height it fell from. How hi… - Perimeter expression evaluated
The perimeter of a certain rectangle is given by the expression 2w + 3w, where w is its width. What is the perimeter whe… - Subtracting a product from a constant
What is the value of 10 − 2p when p = 3? - Positive number whose square doubled is eighteen
For what positive value of x is 2x² equal to 18? - Perimeter of a square from its area
A square has an area of 49 square metres. What is its perimeter, in metres? - Value of an exponential model after one hour
The number of cells in a culture is modelled by N(t) = 10 · 3ᵗ, where t is measured in hours. How many cells does the mo… - Area from a quadratic rule for a rectangle
The area A of a certain rectangle, in square centimetres, is given in terms of its width w, in centimetres, by A = w(w +… - Value of a fraction with a variable
What is the value of 6n/2 when n = 4? - Value of a bracketed expression
What is the value of 2(a + 3) when a = 5? - Side of a square courtyard
A square courtyard has an area of 400 square metres. What is the length of one side, in metres? - Positive value from a square plus a constant
For what positive value of y is y² + 9 equal to 25? - Value of a quadratic model at a point
The distance d, in metres, that a dropped stone falls in t seconds is modelled by d = 5t². How far does the stone fall i… - Bacteria after two doublings
A dish holds 1,000 bacteria, and the number doubles every hour. How many bacteria are in the dish after 2 hours? - Value of a difference of like terms
What is the value of 4c − 2c when c = 3? - Equation with two given solutions
Which of the following equations has both x = 5 and x = −5 as solutions? - Value of a quadratic model
The area A of a certain shape, in square centimetres, is given by A = 2s², where s is a length in centimetres. What is A… - Value after combining three like terms
What is the value of 8d − 3d + d when d = 2? - Nonzero solution of a square equal to a multiple
If x² = 2x and x is not 0, what is the value of x? - Value of a doubling model after one period
The number of lily pads on a pond is modelled by L(w) = 20 · 2ʷ, where w is the number of weeks from now. How many lily … - Value of a linear minus a square term
What is the value of 9w − 2w² when w = 2? - Value of a quadratic expression
What is the value of 3p² − p when p = 4? - Recognising a nonlinear equation
Which of the following equations is NOT a linear equation? - Value of a cubic model
The volume V, in cubic centimetres, of a growing crystal after d days is modelled by V = d³ + 2. What is the volume afte… - Value of a difference of two terms
What is the value of 6k − k² when k = 3? - Positive root after isolating a square
If 10 − x² = 1, what is the positive value of x? - Negative solution of a simple quadratic
If x² + 5 = 30 and x < 0, what is the value of x? - Value that makes a rational expression undefined
For which value of x is the expression 1/(x − 3) undefined? - Negative number whose square is 81
The product of a number and itself is 81. If the number is negative, what is the number? - Vertex from vertex form
The graph of y = (x − 2)² + 5 in the xy-plane is a parabola. What are the coordinates of its vertex? - Next value of a doubling function
The values of an exponential function h are h(0) = 3, h(1) = 6, and h(2) = 12. What is the value of h(3)? - Identifying a decreasing function
Which of the following functions decreases as x increases, for all values of x? - Parabola with vertex at the origin
Which of the following equations represents a parabola whose vertex is at the origin? - Adding two polynomials
Which expression is equivalent to (3x² + 2x) + (x² − 5x)? - Coefficients after distributing
If 5(2x − 1) = ax + b for all values of x, where a and b are constants, what is the value of a − b? - Squaring a substituted expression
If y = 2x, which of the following is equivalent to y² in terms of x? - Square root of a perfect-square monomial
For x > 0, which of the following is equivalent to √(16x²)? - Minimum value of a simple parabola
What is the minimum value of y = x² + 2? - Balance after tripling twice
The balance of an account triples each year. If the balance is 20 dollars now, what will the balance be, in dollars, aft… - Graph that misses the origin
Which of the following functions has a graph that does not pass through the origin? - Side of a square garden from its area
A square garden has an area of 144 square metres. What is the length, in metres, of one side of the garden? - Edge of a cube from its volume
A cube has a volume of 64 cubic units. What is the length of one edge of the cube? - Constant giving two real solutions
The equation x² = k has exactly two real solutions. Which of the following could be the value of k? - Positive fourth root
If x⁴ = 81 and x > 0, what is the value of x? - Distributing and cancelling a term
Which of the following is equivalent to 2(x + y) − 2y? - Adding the same binomial three times
Which of the following is equivalent to (x + 2) + (x + 2) + (x + 2)? - Function whose graph is a parabola
Which of the following functions has a graph that is a parabola? - Combining like terms with a subtraction
Which expression is equivalent to 4x + 3x − 2x? - Negative square root of 49
If x² = 49 and x is negative, what is the value of x? - Identifying a nonlinear function from a table
Which of the following tables gives values of a function that is NOT linear? - Distributing over two variables
Which expression is equivalent to 3(2x + 4y)? - Positive number whose square is 9 less than 90
A positive number, when squared, is 9 less than 90. What is the number? - Function through two given points
Which of the following functions has a graph in the xy-plane that passes through both (1, 5) and (2, 25)? - Adding two equal products
Which expression is equivalent to 2(a + b) + 2(a + b)? - Sum of both square roots
If x² = 36, what is the sum of all possible values of x? - Table matching a square function
Which of the following tables could give values of y = x² for x = 1, 2, and 3? - Subtracting a binomial in brackets
Which expression is equivalent to 8x − (3x + 2)? - Setting up an equation for a square's side
A square has an area of 49 square units. Which of the following equations could be used to find the side length s of the… - Function with the greatest value at x = 3
Which of the following functions has the greatest value when x = 3? - Product of three copies of x
Which of the following is equivalent to x · x · x? - Possible value from a square
If x² = 121, which of the following is a possible value of x? - Dividing a linear expression by a constant
The expression (18x − 12)/6 is equivalent to ax + b for constants a and b. What is the value of a + b? - Product rule for exponents
For x ≠ 0, the expression x⁵ · x³ is equivalent to xⁿ. What is the value of n? - Distributing a monomial
When the expression 2x(x + 5) is written in the form ax² + bx, what is the value of b? - Adding two linear expressions
The sum (x + 7) + (3x − 2) is equivalent to which of the following? - Factoring out the greatest common factor
The expression 6x + 9 is equivalent to k(2x + 3) for a constant k. What is the value of k? - Power of a power
For x ≠ 0, which expression is equivalent to (x²)⁴? - Solving a simple cubic
For what value of x is x³ equal to 64? - Undoing a square root
For what value of x does √x equal 6? - Double root of a squared binomial
What is the only value of x that satisfies (x − 5)² = 0? - Recovering a number from its square, in words
When a positive number is squared and the result is decreased by 11, the outcome is 70. What is the number? - Reading solutions from factored form
What are the solutions to (x − 2)(x + 6) = 0? - Doubling colony in words
A bacteria colony starts with a single cell and doubles in size every hour. After how many hours does the colony first c… - Height of a thrown ball
A ball's height in meters t seconds after being thrown is h(t) = 30t − 5t². What is the ball's height 2 seconds after th… - Daily doubling in context
A video's view count is modeled by v(n) = 10 · 2ⁿ, where n is the number of days since posting. How many views does the … - Opening direction of a parabola
In the xy-plane, which direction does the graph of y = −x² + 5 open? - Lowest point of a lifted parabola
In the xy-plane, the lowest point of the graph of y = x² + 6 is (a, b). What is the value of b? - y-intercept of a quadratic
In the xy-plane, the graph of y = x² − 4x + 9 crosses the y-axis at (0, c). What is the value of c? - Cubing a negative input
The function h is defined by h(x) = x³. What is the value of h(−2)?
Easy–Medium (314)
- Function notation
If f(x) = 2x² − 3, what is f(4)? - Exponent rules
Which expression is equivalent to (2x³)(3x⁴)? - Difference of squares
What are the solutions to x² − 16 = 0? - Expanding a squared binomial
Which expression is equivalent to (x + 5)²? - Simplifying rational expressions
For x ≠ 0, which expression is equivalent to (x² + 3x) / x? - Solving a quadratic equation by taking square roots
What are the solutions to the equation 2(x − 3)² − 32 = 0 ? - Using a known zero of a quadratic to find an unknown coefficient
The function h is defined by h(x) = x² + kx + 9, where k is a constant. If h(3) = 0, what is the value of k ? - Multiplying powers with the same base
Which expression is equivalent to (x³)(x⁴)? - Factoring out a common factor
Which expression is equivalent to 3x² + 12x? - Dividing powers with the same base
For x ≠ 0, which expression is equivalent to x⁶ / x²? - Zeros of a factored quadratic
The function f is defined by f(x) = (x + 2)(x − 6). For which values of x is f(x) = 0? - Interpreting the base of an exponential model
What does the number 1.04 indicate about the population? - Solving x² = k
What are all the solutions to the equation x² = 49? - Subtracting polynomials
Which expression is equivalent to (5x² + 2x) − (3x² − 4x)? - Negative exponent
For x ≠ 0, which expression is equivalent to x⁻³? - Writing an exponential model
A bacteria culture starts with 200 cells, and the number of cells triples every hour. Which function gives the number of… - Where a parabola touches the x-axis
In the xy-plane, the graph of y = (x − 5)² touches the x-axis at exactly one point. What is the x-coordinate of that poi… - Isolating a squared term
If x² + 5 = 30, which of the following gives all possible values of x? - y-intercept of an exponential function
In the xy-plane, the graph of y = 5 · 3ˣ crosses the y-axis at which point? - Positive solution of a quadratic
What is the positive solution to the equation x² − 16 = 0? - Product of conjugates
Which expression is equivalent to (3x + 2)(3x − 2)? - Power of a power then a quotient
For x ≠ 0, which expression is equivalent to (x⁵)² / x⁴? - Growth rate from an exponential model
The size of an ant colony x months from now is modelled by f(x) = 250(1.5)ˣ. By what percent does the colony grow each m… - Not dividing away a solution
A student claims that the only solution to the equation x² = 2x is x = 2. Which of the following correctly describes the… - Solving an exponential function equation
The function g is defined by g(x) = 2ˣ − 3. For what value of x is g(x) = 13? - Multiplying monomials in two variables
Which expression is equivalent to (4a²b)(−3ab³)? - Subtracting fractions with a variable
Which expression is equivalent to 5x/6 − x/3? - Recognising a perfect square trinomial
Which expression is equivalent to x² − 10x + 25? - When a projectile returns to the ground
The height, in feet, of a model rocket t seconds after it is fired from the ground is modelled by h(t) = −16t² + 64t. Ho… - One solution of a factorable quadratic
The equation x² + 3x − 10 = 0 has two solutions. Which of the following is one of them? - Squaring a binomial in two variables
Which expression is equivalent to (x − 3y)²? - Factoring out a negative common factor
Which expression is equivalent to −4x³ + 8x² − 12x? - Isolating x² first
If 5x² = 45, what is the positive value of x? - Difference of squares plus a square
Which expression is equivalent to (a − b)(a + b) + b²? - Axis of symmetry from the x-intercepts
The function g is defined by g(x) = (x − 4)(x + 2). What is the x-coordinate of the vertex of the graph of g in the xy-p… - Square root method, negative solution
If (x + 2)² = 49, what is the negative solution for x? - Factoring a common binomial
Which expression is equivalent to (x + 3)² − (x + 3)(x − 3)? - Recognising decay with a given intercept
Which of the following functions has a graph in the xy-plane with a y-intercept of 7 and represents exponential decay? - Sum of opposite solutions
If (x − 5)(x + 5) = 0, what is the sum of all solutions to the equation? - Square root of an even power
For x > 0, which expression is equivalent to √(x⁶)? - Coefficient of a quadratic from a table
The values in the table satisfy f(x) = ax², where a is a constant. What is the value of a? - Two periods of exponential decay
A population is modelled by P(t) = 500(0.8)ᵗ, where t is the number of years from now. What is the population 2 years fr… - Equation with a single repeated solution
If (x + 3)² = 0, what is the value of x? - Function from its x-intercepts
Which of the following functions has a graph in the xy-plane with x-intercepts at x = −1 and x = 4? - Cube root of a negative number
If x³ = −8, what is the value of x? - Counting solutions of a perfect-square equation
How many distinct real solutions does the equation x² + 6x + 9 = 0 have? - Spotting the nonlinear function
Which of the following functions is NOT a linear function? - Square root of a multiple of x
If √(2x) = 4, what is the value of x? - Sum of the roots of a factorable quadratic
If x² − 7x + 12 = 0, what is the sum of the two solutions to the equation? - Squaring a monomial then multiplying
Which expression is equivalent to (3x)² · x? - Percent change per step of an exponential function
The function g is defined by g(t) = 200(0.6)ᵗ. By what percent does g(t) change from one whole-number value of t to the … - A solution of x² = x
Which of the following is a solution to the equation x² = x? - Power of a power divided by a power
For x ≠ 0, which expression is equivalent to (x²)³ / x⁴? - Table that could show a linear function
Which of the following tables of values could represent a LINEAR function? - Product of the two square roots
The equation x² − 25 = 0 has two solutions. What is the product of the two solutions? - Multiplying with a negative exponent
For x ≠ 0, which expression is equivalent to x⁻² · x⁵? - Recognising a downward parabola
Which of the following functions has a graph in the xy-plane that is a parabola opening DOWNWARD? - Positive root of a trinomial
What is the positive value of x that satisfies x² − 3x − 10 = 0? - Expanding a product with mixed signs
Which expression is equivalent to (x + 4)(x − 2)? - Length of a rectangle from area and width
A rectangle has an area of 6x³ − 9x² square units and a width of 3x² units, where x > 2. Which expression represents the… - Negative root of a quadratic
What is the negative solution to the equation x² + 3x − 4 = 0? - Evaluating an exponential function
The function f is defined by f(x) = 2ˣ + 3. What is the value of f(5)? - Area of a rectangle as a product of binomials
A rectangle has length x + 4 metres and width x + 2 metres. Which expression represents its area, in square metres? - Bacteria tripling every day
A colony of 2 bacteria triples in number every day. How many bacteria will there be after 3 days? - Number from a ratio condition
The ratio of twice a number to two less than the number is 4 to 1. What is the number? - Length of a rectangle from area and width (quadratic area)
A rectangle has an area of x² + 5x + 6 square units and a width of x + 2 units, where x > 0. What is the length of the r… - Graph of an exponential function: which is true
The function f is defined by f(x) = 3(2)ˣ. Which statement about the graph of y = f(x) in the xy-plane is true? - Rational equation from a work context
Working alone, one pump can empty a tank in 2 hours and a second pump can empty it in 3 hours. The time t, in hours, for… - Sum of coefficients of a squared binomial
When (2x + 1)² is written in the form ax² + bx + c, what is the value of a + b + c? - Recognising a quadratic from its description
A rectangular pen is to be built against a wall using 40 metres of fencing for the three open sides. If the side perpend… - Solving an equation with a squared term
If 2x² + 1 = 19 and x > 0, what is the value of x? - Simplifying a monomial fraction in context
The area of a rectangle is 2x² square units and its length is 6x units, where x > 0. If x = 9, what is the width of the … - Evaluating a quadratic model in context
A market stall's daily profit P, in dollars, from selling n crates of apples is modelled by P(n) = 3n² − 5. What is the … - Solving a quadratic from a formula in context
The distance d, in metres, a dropped object falls in t seconds is approximately d = 5t². How many seconds does it take a… - Simplifying a product plus a constant
Which expression is equivalent to (x + 6)(x − 6) + 36? - Exponential growth: value after a whole number of periods
A rumour is heard by 4 people on the first day, and the number who have heard it triples each day. How many people have … - Positive root of a quadratic in a simple frame
The square of a positive number, minus the number, is 30. What is the number? - Exponent from a product of powers
For x > 0, the expression x² · (x³)² is equal to xⁿ, where n is a constant. What is the value of n? - Doubling repeatedly
A single bacterium culture starts with 20 cells and doubles in number every hour. How many cells are there after 4 hours… - Rectangle dimensions from area and a relation
A rectangle is 6 centimetres longer than it is wide, and its area is 40 square centimetres. What is its width, in centim… - Simplifying a monomial division in context
A garden bed has an area of 3x² + 6x square metres and a width of 3x metres, where x > 0. What is its length, in metres,… - Identifying exponential growth from a table
The table shows values of a function f. Which equation could define f? - Number from a quarter of its square
A positive number is such that one quarter of its square equals 9. What is the number? - Exponent from powers and a quotient
For x > 0, (x⁴)² / x⁵ = xⁿ. What is the value of n? - Numerator after adding fractions with the same denominator
For x ≠ 0, the sum 2/x + 3/x is equal to k/x, where k is a constant. What is the value of k? - Recognising exponential decay from a description
A pond loses 20% of its water volume each week through evaporation with no inflow. Which type of function best models th… - Equation with a squared binomial
If (x + 4)² = 49 and x > 0, what is the value of x? - Combining like terms after distributing
Which expression is equivalent to 2(x + 4) − 3(x − 1)? - Growth by 50% twice
A puppy weighed 12 kilograms at the start of a year and its weight grew by 50% in each of the next two half-years. What … - Equation with a squared term and a constant
If 3x² − 5 = 43 and x > 0, what is the value of x? - Cubing a monomial
Which expression is equivalent to (2x²)³? - Recognising a linear versus exponential situation
A savings account adds $50 to its balance every month with no interest. Which type of function models the balance over t… - Equation with x squared plus a constant
If x² + 3 = 52 and x > 0, what is the value of x? - Coefficient after squaring and dividing
For x ≠ 0, (4x)²/(2x) = kx, where k is a constant. What is the value of k? - Choosing the exponential model
A cup of coffee contains 200 milligrams of caffeine. The amount in the body falls by half every 5 hours. Which function … - Quadratic equation from a formula
The kinetic energy E of a cart, in joules, is given by E = 2v² − 8, where v is its speed in metres per second and v > 2.… - Exponent after a quotient is squared
For x > 0, (x⁵/x²)² = xⁿ. What is the value of n? - Recognising exponential from a description
The number of views of a video triples every day. Which type of function models the number of views over time? - Equation with a squared term, from a formula
The energy E stored in a spring is given by E = 2x² − 8, where x is its extension in centimetres and x > 2. What is x wh… - Value of a ratio after substitution
If y = 2x and x ≠ 0, what is the value of (x + y)/x? - Recognising exponential decay from a description
A cup of tea's temperature above room temperature falls by 10% every minute. Which type of function best models the temp… - Side of a patio from a tile count
The number of tiles needed for a square patio of side s metres is given by 3s² + 2, which includes 2 spare tiles. If 50 … - Width of a rectangle from area and length
A rectangle has an area of x² + 3x square metres and a length of x + 3 metres, where x > 0. If x = 6, what is the rectan… - Choosing the exponential growth model
A colony of 80 bacteria doubles every 3 hours. Which function gives the population P after t hours? - A number times twice itself
The product of a positive number and twice that number is 98. What is the number? - Difference of squares from a sum and a difference
The sum of two numbers is 9 and their difference is 3. What is the difference of the squares of the two numbers? - Solving an exponential equation by matching bases
If 3^(2x − 1) = 81, what is the value of x? - x-intercept of a shifted exponential graph
In the xy-plane, the graph of y = 2^x − 8 crosses the x-axis at one point. What is the x-coordinate of that point? - y-intercept after a vertical shift
The graph of y = x² − 6x + 5 in the xy-plane is shifted 3 units upward. What is the y-intercept of the resulting graph? - Growth factor over several doubling periods
A population doubles every 5 years. By what factor does the population grow over 20 years? - Percent change between consecutive outputs of a decay model
The function p is defined by p(t) = 500(0.8)ᵗ. By what percent does p decrease from t = 1 to t = 2? - Identifying a parabola from its vertex and direction
Which of the following equations has a graph in the xy-plane that is a parabola with vertex (−1, 4) opening upward? - Least value in the domain of a square-root function
The function h is defined by h(x) = √(x − 2). What is the least value of x in the domain of h? - Difference of a cubic function at opposite inputs
The function f is defined by f(x) = x³. What is the value of f(2) − f(−2)? - Input of an exponential function from its output
In the xy-plane, the graph of y = 2ˣ passes through the point (a, 32). What is the value of a? - Value that makes a rational expression undefined
For what positive value of x is the expression (x − 4)/(x² − 16) undefined? - Two numbers from a ratio and a product
One positive number is 3 times another, and their product is 48. What is the larger number? - True statement about a parabola in vertex form
Which of the following statements is true about the graph of f(x) = (x − 2)² + 3 in the xy-plane? - Constant for a circle through a given point
In the xy-plane, the circle x² + y² = c passes through the point (3, −4), where c is a constant. What is the value of c? - Value of x² from a difference-of-squares equation
If (x − 3)(x + 3) = 27, what is the value of x²? - Product of two powers with the same base
If x = 2ᵃ and y = 2ᵇ, which expression is equal to xy? - Cube of a negative fourth root
If x⁴ = 16 and x < 0, what is the value of x³? - Value of a multiple of a known expression
If x + 2y = 5, what is the value of 3x + 6y − 4? - Number of terms in an expanded product
When x(x + 1)(x + 2) is expanded and like terms are combined, how many terms does the resulting polynomial have? - Time for a falling object to cover a distance
A stone dropped from rest falls a distance of d = 5t² metres in t seconds. How many seconds does it take to fall 80 metr… - y-intercept after a vertical stretch
The graph of y = 2ˣ in the xy-plane is stretched vertically by a factor of 3. What is the y-intercept of the new graph? - Equation of a shifted parabola
The graph of y = x² in the xy-plane is shifted 2 units to the left and 5 units down. Which of the following is an equati… - Interpreting the base of an exponential model
The value of an investment n years after it is made is modelled by V(n) = 1,000(1.05)ⁿ dollars. What is the best interpr… - Negative exponent of a fraction
What is the value of (1/2)⁻³? - Evaluating a polynomial in 2x
If x = 1/2, what is the value of (2x)³ − (2x)² + 2x? - Sum of exponents from two separate exponential equations
If 3ˣ = 81 and 2ʸ = 32, what is the value of x + y? - Difference of two values of a square-root function
The function h is defined by h(x) = √(2x + 1). What is the value of h(12) − h(4)? - Cross term from the square of a sum
If x = 5 and y = 2, what is the value of (x + y)² − (x² + y²)? - Counting the values where a rational expression is undefined
For how many real values of x is the expression (x + 2)/(x² − 4) undefined? - Solving a linear equation in a square root
If 3√x = 12, what is the value of x? - Ratio of sides from a ratio of cube volumes
The volume of one cube is 8 times the volume of a smaller cube. The side length of the larger cube is how many times the… - Axis of symmetry of a parabola in standard form
In the xy-plane, the graph of f(x) = 3x² − 12x + 7 is symmetric about the vertical line x = k. What is the value of k? - Numeric difference of squares
What is the value of 99² − 98²? - Ratio through a chain of proportions
If a = 2b and b = 3c, where c ≠ 0, what is the value of a/c? - Input giving a fractional output of an exponential
The function f is defined by f(x) = 2ˣ. For what value of x is f(x) = 1/8? - Product of powers of 2 with a negative exponent
What is the value of 2⁵ · 2⁻³ · 2²? - Negative exponent from a decimal
If 10ˣ = 0.001, what is the value of x? - Reflection of an exponential graph across the y-axis
In the xy-plane, which of the following is an equation of the reflection of the graph of y = 2ˣ across the y-axis? - Point on the graph of a shifted exponential
Which of the following points lies on the graph of f(x) = 3ˣ + 1 in the xy-plane? - Vertex of a shifted absolute-value graph
In the xy-plane, the graph of f(x) = |x − 3| is shifted 2 units to the right. What is the x-coordinate of the vertex of … - Sum of two values of a polynomial
The function p is defined by p(x) = x⁴ − x + 3. What is the value of p(0) + p(1)? - Evaluating after choosing the negative root
If 2x² = 50 and x < 0, what is the value of x + 1? - Value after a fixed number of tripling periods
The number of bacteria in a dish triples every 4 hours. If there are 200 bacteria at noon, how many are there at 8 p.m. … - Least value in the range of a quadratic
The function f is defined by f(x) = x² + 1 for all real x. What is the least value in the range of f? - Sum of all solutions of a quadratic with a zero root
If x² = 7x, what is the sum of all solutions of the equation? - Positive root of a factorable quadratic
The equation x² − 5x − 6 = 0 has one positive solution. What is it? - Square-root equation
If √(2x + 9) = 5, what is the value of x? - Width of a garden twice as long as it is wide
A rectangular garden is twice as long as it is wide, and its area is 32 square metres. What is the width of the garden, … - Sum of solutions of a shifted square
The equation (x − 1)² = 4 has two solutions. What is their sum? - Intersection of a parabola and a line
In the xy-plane, the graphs of y = x² and y = 2x + 3 intersect at two points. What is the x-coordinate of the point with… - Both solutions of a square-root form
Which of the following lists all real solutions of x² − 25 = 0? - Exponential equation with a shifted exponent
If 2^(x − 2) = 16, what is the value of x? - Middle coefficient of a rectangle's area
A rectangle has length 2x + 3 and width x − 1. Its area can be written as 2x² + bx − 3, where b is a constant. What is t… - Constant after simplifying a difference
When 3(x − 2) − 2(x + 1) is simplified, the result is x + c, where c is a constant. What is the value of c? - Factor of a difference of squares
The expression x² − 16 can be written as (x + 4)(x − k), where k is a positive constant. What is the value of k? - Simplifying a quotient of monomials
For x ≠ 0, the expression 12x³/(4x) simplifies to kxⁿ, where k and n are constants. What is the value of k + n? - Adding fractions with a variable denominator
For x ≠ 0, the sum 1/x + 1/(2x) can be written as k/(2x), where k is a constant. What is the value of k? - Repeated root of a perfect-square quadratic
The equation x² + 6x + 9 = 0 has exactly one solution. What is it? - Number from its square root plus two
Adding 2 to the square root of a positive number gives 7. What is the number? - Positive input giving a target output
For what positive value of x does x² − 4 equal 5? - Largest root of a factorable cubic
The equation x³ − 4x = 0 has three solutions. What is the largest of them? - Equation with two given solutions
Which of the following equations has solutions x = 3 and x = −5? - Difference in profit from a quadratic model
A company's profit, in thousands of dollars, from selling x thousand units is modelled by P(x) = x² − 2x. How much great… - Vertex form from a lowest point
The graph of a quadratic function in the xy-plane has its lowest point at (2, −3). Which equation could define the funct… - Years for a colony to reach a size
A colony of bats has 5 members, and its size after x years is modelled by 5·2ˣ. After how many years will the colony hav… - Behaviour of an exponential as x decreases
In the xy-plane, which of the following describes the graph of y = 2ˣ as x decreases without bound? - Middle coefficient of a squared binomial
When (x + 5)² is expanded and written as x² + bx + 25, what is the value of b? - Dividing a binomial by a monomial
For x ≠ 0, the expression (6x² + 9x)/(3x) simplifies to ax + b, where a and b are constants. What is the value of a + b? - Second factor of a trinomial
For all values of x, x² − 7x + 12 = (x − 3)(x − k), where k is a constant. What is the value of k? - Coefficient of x after distributing 2x
The expression 2x(3x − 4) is equivalent to 6x² + kx, where k is a constant. What is the value of k? - Exponent after dividing powers
For x ≠ 0, the expression x⁷/x⁴ is equal to xⁿ. What is the value of n? - Constant after combining two distributed terms
The expression 4(x + 2) − 2(x − 3) simplifies to 2x + c, where c is a constant. What is the value of c? - Product of the roots of a difference of squares
The equation x² − 16 = 0 has two solutions. What is their product? - Positive root from a word equation
Twice the square of a positive number, minus the number itself, is 15. What is the number? - Solving with a fractional exponent
If x^(3/2) = 8 and x > 0, what is the value of x? - Larger root of a quadratic in standard form
The equation x² − x − 20 = 0 has two solutions. What is the larger one? - System with a parabola and a horizontal line
How many solutions does the system y = x² − 3 and y = 1 have? - Exponent that makes 4 to the x equal 8
For what value of x does 4ˣ equal 8? - Missing table value of an exponential-plus-constant function
The function p is defined by p(x) = 3·2ˣ + 1. The table shows some of its values. x | p(x) 0 | 4 1 | 7 2 | ? What is t… - Effect of a horizontal shift on a vertex
The graph of y = (x − 4)² is the graph of y = x² shifted in which way? - Evaluating a quadratic with two terms
The function g is defined by g(x) = 2x² − 3x. What is the value of g(3)? - Vertex of a parabola from its factored form
In the xy-plane, the graph of y = (x − 1)(x − 5) is a parabola. What is the x-coordinate of its vertex? - Amount after two half-lives
A sample of a substance has a mass of 200 grams. Its mass halves every 3 days. What is its mass, in grams, after 6 days? - Reading a maximum from vertex form
The height h, in metres, of a ball t seconds after it is thrown is given by h(t) = −5(t − 2)² + 25. What is the greatest… - Coefficient of x in an expanded product
What is the coefficient of x when (x + 4)(x − 1) is expanded? - Simplifying a product minus a term
Which of the following is equivalent to x(x + 3) − 3x? - Ratio from a substitution
If a = 2b and b ≠ 0, what is the value of (a + b)/b? - Cubing a monomial
Which of the following is equivalent to (2x)³? - Difference between a squared sum and a sum of squares
Which of the following is equivalent to (x + y)² − (x² + y²)? - Positive input for a squared binomial
For what positive value of x is (x + 1)² equal to 16? - Number whose square exceeds it by 30
The square of a positive number is 30 more than the number itself. What is the number? - Intersection of an exponential and a horizontal line
In the xy-plane, the graphs of y = 2ˣ and y = 16 intersect at the point (a, 16). What is the value of a? - Larger of two consecutive integers with a given product
The product of two consecutive positive integers is 56. What is the larger of the two integers? - Positive number whose root is half of it
For what positive value of x is √x equal to x/2? - Two inputs with the same output for a shifted square
The function f is defined by f(x) = (x − 3)² + 1. If f(5) = f(a) and a ≠ 5, what is the value of a? - Time for a dropped stone to land
A stone is dropped from a cliff. Its height h, in metres, above the ground t seconds later is given by h(t) = 80 − 5t². … - Equation for a table that doubles the square
The table shows values of a function. x | y 1 | 2 2 | 8 3 | 18 Which equation describes y in terms of x? - Car value after one year of depreciation
The value V, in dollars, of a car t years after purchase is modelled by V(t) = 20,000(0.8)ᵗ. What is the value of V(1)? - Factor by which a square's area grows
The area of a square with side length s is given by A(s) = s². By what factor does the area increase when the side lengt… - Recognising an exponential function
Which of the following functions is exponential? - Positive x-intercept of a lowered parabola
In the xy-plane, the graph of y = x² is shifted 4 units down. What is the positive x-intercept of the new graph? - Counting zeros of a factored function
The function f is defined by f(x) = x(x − 4). For how many values of x is f(x) = 0? - Positive input where a square equals a double
The functions f and g are defined by f(x) = x² and g(x) = 2x. For what positive value of x is f(x) equal to g(x)? - Extra customers between two days of doubling
In its first week, a new shop's number of customers doubled every day: 5 on day 1, 10 on day 2, and so on. How many more… - Simplifying with two variables
Which of the following is equivalent to 3(a + b) − 2b? - Coefficient after squaring a monomial
When (4x)² is written in the form kx², what is the value of k? - Sum of three consecutive integers
The smallest of three consecutive integers is n. Which expression gives the sum of the three integers? - Scaling a known expression
If 2x − y = 5, what is the value of 4x − 2y? - Sum of two numbers from a difference-of-squares fact
Two positive numbers differ by 2, and the difference of their squares is 20. What is the sum of the two numbers? - Positive number from its square plus itself
A positive number added to its own square gives 30. What is the number? - Meeting point of two curves
In the xy-plane, the graphs of y = x² and y = x³ intersect at (0, 0) and at one other point. What is the x-coordinate of… - Folds of paper from a layer count
A sheet of paper has 2ⁿ layers after it has been folded in half n times. How many times must the sheet be folded to have… - Equation with a square root and a constant
Which of the following values of x satisfies √(3x + 6) = 3? - Coefficient left after subtracting a sum of squares
For all values of x, (x + 3)² − (x² + 9) = kx, where k is a constant. What is the value of k? - Sum of coefficients after simplifying
When the expression 3(2x − 5) − 4x is written in the form ax + b, where a and b are constants, what is the value of a + … - Coefficient and exponent of a product of monomials
The product 2x² · 5x³ can be written as kxⁿ, where k and n are constants. What is the value of k + n? - Distance between x-intercepts of a parabola
In the xy-plane, the parabola y = x² − 4 meets the x-axis at two points. How far apart are those two points? - Smaller root of a shifted square
If (x − 3)² = 16, what is the smaller of the two possible values of x? - Increase of an exponential over one unit
By how much does the value of 2ˣ increase when x changes from 4 to 5? - Least value of a shifted square
What is the least possible value of the expression (x − 2)² + 5 for real values of x? - Coefficient of x in a product of binomials
When (3x + 2)(x − 4) is multiplied out and written as a polynomial, what is the coefficient of the x term? - Product of coefficient and exponent after dividing monomials
The expression 12x⁵ / (4x²) simplifies to axⁿ, where a and n are constants. What is the value of a·n? - Coefficient from a known root
The equation x² + bx + 12 = 0 has x = 3 as one of its solutions, where b is a constant. What is the value of b? - Positive value where a square equals a linear expression
For what value of x greater than 0 is x² equal to 5x + 14? - Profit from a quadratic model
A workshop's daily profit, in dollars, from making x chairs is given by P(x) = −x² + 20x − 36. What is the profit, in do… - Value approached by an exponential function
The function f is defined by f(x) = 2ˣ + 3. As x decreases without bound, the value of f(x) gets closer and closer to wh… - Constant term of a squared binomial
When (x − 3)² is written in the form x² + bx + c, what is the value of c − b? - Constant after expanding and combining
When 4(x + 2) − 2(x − 1) is simplified, what is the constant term? - Increase in a square's side from its areas
The area of a square garden is 25 square metres. Each side is lengthened by the same amount, and the new area is 49 squa… - Positive solution of a scaled square
If 2x² − 18 = 0, what is the positive value of x? - Two triplings
A quantity starts at 100 and triples every hour. What is the quantity after 2 hours? - Coefficient after distributing and combining
When 5(x − 2) + 3x is simplified, what is the coefficient of x? - Sum of the non-leading coefficients of a squared binomial
When (x + 4)² is written as x² + bx + c, what is the value of b + c? - Gap between the two solutions of a square equation
The equation x² = 9 has two solutions. How far apart are they on the number line? - Amount after three doublings
A sum of $5 is doubled three times. What is the result, in dollars? - Constant after subtracting a binomial
When 2(3x + 1) − (x − 4) is simplified, what is the constant term? - Coefficient from a monomial times a binomial
When x(x + 7) is written as x² + kx, what is the value of k? - Positive solution of a squared linear expression
If (2x − 1)² = 25 and x > 0, what is the value of x? - Smaller of two numbers from product and difference
Two positive numbers differ by 2 and have a product of 48. What is the smaller number? - Input where a quadratic reaches a value
The function p is defined by p(x) = x² − 4. For what positive value of x is p(x) = 5? - One step of 50% growth
A quantity grows by 50% each year. If it is 80 now, what will it be in one year? - Coefficient of a sum of like square terms
The sum x² + 3x² + 5x² is equivalent to kx², where k is a constant. What is the value of k? - Number from a square and a linear clue
The square of a positive number is 19 more than twice the number minus 4. What is the number? - Increase over one step of an exponential model
The value of an investment after t years is modelled by V(t) = 3 · 2ᵗ thousand dollars. By how many thousand dollars doe… - Larger root of a product equal to 50
The equation (x − 2)(x + 3) = 50 has two solutions. What is the larger one? - Positive input giving a specific output
The function f is defined by f(x) = x² − 3x. For what positive value of x is f(x) = 4? - Population after two triplings
A population of 200 insects triples every week. What is the population after 2 weeks? - Coefficient and constant after two distributions
When 3(2x − 1) + 2(x + 4) is written as ax + b, what is the value of a − b? - Constant in a product of conjugates
When (x − 6)(x + 6) is written as x² − c, what is the value of c? - Sum of the solutions of a factored product
The equation (x + 1)(x − 3) = 5 has two solutions. What is their sum? - Positive number from a square minus a multiple
The square of a positive number, minus twice the number, equals 15. What is the number? - Increase of a quadratic function over an interval
The function f is defined by f(x) = 3x² + 1. By how much does f(x) increase as x goes from 1 to 2? - Three doublings from 50
A number of bacteria doubles every hour. If there are 50 now, how many will there be in 3 hours? - Expression that simplifies to a constant
For all values of x, 4(x − 1) − 2(2x − 5) is equal to which constant? - Constant from a conjugate product
When (x + 7)(x − 7) is written as x² − k, what is the value of k? - Constant that makes a given value a solution
For what value of k does the equation x² + k = 0 have x = 4 as a solution? - Positive number from a square minus one
The square of a positive number, decreased by 1, equals 8. What is the number? - Two doublings of a deposit
A deposit of $250 doubles in value every 5 years. What will it be worth, in dollars, after 10 years? - Value of a quadratic rule at 3
A rule assigns to each number x the value 2x² − 5. What value does the rule assign to 3? - Product of a sum and a difference
When (x + 8)(x − 8) is written as x² − c, what is the value of c? - Negative solution of a quadratic equation
For what negative value of x is x² + 2x equal to 8? - Value satisfying a quadratic and a linear condition
Which value of x satisfies both x² = 16 and x + 3 = 7? - Value of a quadratic function at 2
The function w is defined by w(x) = 5x² − 3. What is the value of w(2)? - Constant term after two distributions
When 7(x − 2) − 3(x − 4) is simplified, what is the constant term? - Area of a rectangle as a polynomial
A rectangle has a length of 3x + 2 units and a width of x units. Which expression represents the area of the rectangle, … - Number from its square plus itself
The square of a positive number, plus the number itself, is 42. What is the number? - Value of a quadratic rule at 3
The function d is defined by d(x) = 3x² + 2x. What is the value of d(3)? - Side of a square from an area relation
The area of a square, in square units, is 10 more than three times its side length. What is the side length of the squar… - System with no real solution
Which of the following systems of equations has no real solution? - Negative solution of an absolute-value equation
If |x − 4| = 6 and x < 0, what is the value of x? - Expanding then subtracting a square term
Which expression is equivalent to 2x(3x − 4) − x²? - Constant in a squared binomial
If (x − a)² = x² − 14x + 49 for all values of x, where a is a positive constant, what is the value of a? - Power of a power over a power
For y ≠ 0, which expression is equivalent to (y³)² / y⁴? - Function for a growing population
A town's population is 8,000 and grows by 5 percent each year. Which function gives the population P after t years? - Growth factor over several hours
A bacterial culture grows by a factor of 4 every hour. By what factor does the culture grow over 3 hours? - Reading a parabola's vertex
In the xy-plane, the graph of y = f(x) is a parabola that opens upward with its vertex at (2, −3). Which of the followin… - Value of a car after two years of loss
A car bought for 20,000 dollars loses 20 percent of its value each year. What is its value, in dollars, after 2 years? - Square of a binomial minus a square
Which expression is equivalent to (x + 3)² − x²? - Solutions of a quadratic in a word setting
A number squared, plus the number, equals 12. Which of the following could be the number? - Doubling three times
A colony of bacteria doubles in size every hour. If the colony has 100 bacteria at noon, how many bacteria will it have … - Product of coefficients after combining
If 2(x + y) + 3(x − y) = ax + by for all values of x and y, where a and b are constants, what is the value of a·b? - Positive solution of a cubic minus a linear term
If x³ − x = 0 and x > 0, what is the value of x? - Y-intercept of a factored quadratic
In the xy-plane, the graph of y = (x + 1)(x − 3) crosses the y-axis at what value of y? - Multiplying two monomials
Which expression is equivalent to (5x)(2x²)? - Positive solution of a factored equation
If (x + 4)(2x − 1) = 0 and x > 0, what is the value of x? - Input giving the minimum of a factored quadratic
The function g is defined by g(x) = (x − 4)(x + 2). At what value of x does g reach its minimum value? - Adding two fractions of x
Which expression is equivalent to x/2 + x/3? - The other solution of a quadratic
The equations x² − 9 = 0 and x − 3 = 0 share one solution. What is the other solution of x² − 9 = 0? - Describing a horizontal shift of a parabola
In the xy-plane, how is the graph of y = (x − 3)² related to the graph of y = x²? - Constant inside a factored form
If 6x + 9 = 3(2x + k) for all values of x, what is the value of k? - Equation with exactly one real solution
Which of the following equations has exactly one real solution? - Constant term of a conjugate product
When (x + 6)(x − 6) is expanded and written as x² + c, what is the value of c? - Difference of two squared binomials
For every value of x, (4x − 1)² − (4x + 1)² = kx. What is the value of k? - Gap between the roots' constants
The expression x² + 10x + 21 factors as (x + a)(x + b), where a and b are constants with a < b. What is the value of b −… - Factor completely: constant multiplier
The expression 5x² − 20 is equivalent to k(x − 2)(x + 2) for a constant k. What is the value of k? - Cubing a monomial
The expression (2x³)³ is equivalent to a·x⁹ for a constant a. What is the value of a? - Coefficient and exponent after dividing monomials
For x ≠ 0, the quotient 12x⁵ ÷ 4x² equals c·xⁿ for constants c and n. What is the value of c + n? - Counting distinct solutions of a perfect square
How many distinct real solutions does x² − 10x + 25 = 0 have? - Hidden perfect square in a quadratic
If x² − 12x + 36 = 25 and x < 6, what is the value of x? - Equation with a radical
For what value of x does √(2x + 7) equal 5? - Exponential equation with shifted exponent
For what value of x does 3ˣ⁻² equal 27? - Quadratic with x on both sides
If x² = 5x and x ≠ 0, what is the value of x? - x-intercepts of a parabola
In the xy-plane, at which points does the graph of y = x² − 9 cross the x-axis? - Vertex x from vertex form
The graph of y = (x − 4)² + 2 has its vertex at the point (a, b). What is the value of a? - Coefficient of an exponential from one value
The function g is defined by g(x) = c · 3ˣ for a constant c, and g(2) = 45. What is the value of c? - Growth by equal factors
A quantity is modeled by a function that increases by the same FACTOR over every interval of equal length. Which kind of… - Sum of x-intercepts from factored form
In the xy-plane, the graph of y = (x − 1)(x + 7) has two x-intercepts. What is the sum of their x-coordinates? - Smaller zero of a quadratic
The function f(x) = x² − 6x + 8 has two zeros. What is the smaller of the two? - Evaluating exponential growth in context
A culture's population is modeled by P(t) = 500 · 3ᵗ, with t in days. What is the population at t = 2?
Medium (315)
- Quadratic equations
What are the solutions to x² − 5x + 6 = 0? - Vertex form of a parabola
The graph of f(x) = (x − 3)² + 4 is a parabola. What are the coordinates of its vertex? - Factoring polynomials
If x² − 2x − 8 = (x + 2)(x − a), what is the value of a? - Radical equations
If √(x + 5) = 4, what is the value of x? - Composition of functions
If f(x) = x + 3 and g(x) = 2x, what is f(g(4))? - Solving quadratics by factoring
What are the solutions to x² + 6x + 8 = 0? - Systems with a nonlinear equation
The system y = x² and y = x + 6 has two solutions. What is the sum of the two x-values of those solutions? - Squaring a binomial
Which expression is equivalent to (2x − 3)²? - Factoring a trinomial
Which expression is equivalent to x² + 5x − 14? - Multiplying powers with fractional exponents
For x > 0, which expression is equivalent to x^(1/2) · x^(3/2)? - Simplifying a square root
Which expression is equivalent to √50? - y-intercept of a quadratic
The function f is defined by f(x) = 2x² − 8x + 3. What is the y-coordinate of the y-intercept of the graph of f in the x… - x-coordinate of the vertex from standard form
The function f is defined by f(x) = x² − 6x + 1. What is the x-coordinate of the vertex of the graph of f? - Solving a quadratic by factoring
What are the solutions to the equation x² − 5x + 6 = 0? - Greatest common factor with two variables
Which expression is equivalent to 6x²y − 9xy²? - Difference of squares with a coefficient
Which expression is equivalent to 4x² − 25? - Difference of two squared binomials
Which expression is equivalent to (x + 4)² − (x − 4)²? - Axis of symmetry from the zeros
The graph of a quadratic function in the xy-plane has x-intercepts at (−1, 0) and (7, 0). What is the equation of the gr… - Half-life
A sample contains 80 grams of a substance that decays so that its mass halves every 5 years. How many grams remain after… - Quadratic formula
Which of the following gives all solutions to the equation x² + 2x − 4 = 0? - Minimum value from factored form
The function f is defined by f(x) = (x − 2)(x + 6). What is the minimum value of f(x)? - Interpreting a decay factor
Which of the following best describes how the population changes from one year to the next? - Finding a coefficient from a known solution
In the equation x² + kx + 12 = 0, k is a constant. If x = 3 is a solution, what is the value of k? - Factoring by grouping
Which expression is equivalent to x³ + 2x² + 3x + 6? - Factoring a trinomial with a leading coefficient
Which expression is equivalent to 2x² + 7x + 3? - Cube root of a monomial
Which expression is equivalent to ∛(27x⁶)? - Distance between the x-intercepts
In the xy-plane, the graph of y = x² − 2x − 15 crosses the x-axis at two points. What is the distance between those two … - Sum of solutions of a squared binomial
If (x + 1)² = 36, what is the sum of all solutions to the equation? - Rational equation with the variable in both places
If x/(x − 2) = 3 and x ≠ 2, what is the value of x? - Adding fractions with different powers of x
For x ≠ 0, which expression is equivalent to 3/x + 2/x²? - Expanding a difference of cubes
Which expression is equivalent to (x − 2)(x² + 2x + 4)? - Decay rate from an exponential model
The value of a car, in dollars, t years after it is bought is modelled by V(t) = 12,000(0.85)ᵗ. By what percent does the… - Squares that cancel
If x² + 4x = x² + 12, what is the value of x? - Isolating a square root before squaring
What value of x satisfies 2√x − 3 = 7? - Power of a power with a negative exponent
For x ≠ 0, which expression is equivalent to (x²)³ · x⁻²? - Subtracting square roots
Which expression is equivalent to √72 − √18? - Matching a description to an equation
Which of the following equations has a graph in the xy-plane that opens downward and has its vertex on the x-axis? - Exponential function from two values
The function g is defined by g(x) = a · bˣ, where a and b are positive constants. If g(0) = 5 and g(1) = 15, what is the… - Parabola and a line below it
The parabola y = x² and the horizontal line y = −1 are graphed in the xy-plane. At how many points do the two graphs int… - Dividing monomials in two variables
For x ≠ 0 and y ≠ 0, which expression is equivalent to 12x³y² / (4xy⁵)? - Compound growth over two periods
An account balance grows by 5% each year. If the balance is $2,000 now, what will it be, in dollars, after 2 years? - Exponent equation with a linear exponent
If 3^(2x − 1) = 2,187, what is the value of x? (Note: 2,187 = 3⁷.) - Fractional exponent raised to a power
For x > 0, which expression is equivalent to (x^(1/3))⁶? - Counting distinct x-intercepts
The function f is defined by f(x) = (x − 2)²(x + 1). At how many distinct points does the graph of f cross or touch the … - Quadratic from a product of related numbers
The product of a positive number and the number that is 4 less than it is 21. What is the positive number? - Expanding and cancelling a squared term
Which expression is equivalent to 2(x + 3)² − 2x²? - Isolating the cross term
Which expression is equivalent to (a + b)² − (a² + b²)? - Minimum value of a quadratic with a leading coefficient
The function f is defined by f(x) = 3x² + 12x − 1. What is the minimum value of f(x)? - Difference of two squared binomials with a coefficient
Which expression is equivalent to (2x + 5)² − (2x − 5)²? - Model for a quantity that doubles every 6 hours
A culture of yeast cells doubles in size every 6 hours. If it starts with 5 million cells, which function gives the numb… - Isolating the power first
If 5 · 2ˣ = 80, what is the value of x? - Simplifying by factoring both parts
For x ≠ 0 and x ≠ −3, which expression is equivalent to (x² − 9)/(x² + 3x)? - Three years of 10% decline
A machine's value falls by 10% each year. If it is worth $900 now, what will it be worth, in dollars, in 3 years? - Exponential equation with a fractional result
If 4ˣ = 1/16, what is the value of x? - Factoring completely with a difference of squares
Which expression is equivalent to x³ − x? - Larger zero of a quadratic
The function f is defined by f(x) = x² − 4x − 12. The function has two zeros. What is the larger zero? - Exponent that is a multiple of x
If 2^(2x) = 64, what is the value of x? - Adding fractional coefficients
Which expression is equivalent to (1/2)x + (1/3)x? - y-intercept from vertex form
In the xy-plane, the graph of y = 3(x − 2)² − 5 crosses the y-axis at the point (0, k). What is the value of k? - Hidden perfect square
If x² + 9 = 6x, what is the value of x? - Factoring out a common factor first
Which expression is equivalent to 3x² + 6x − 9? - Counting x-intercepts of a raised parabola
At how many points does the graph of y = (x − 3)² + 1 cross the x-axis in the xy-plane? - Matching exponents in a product
If 2ˣ · 2³ = 2⁷, what is the value of x? - Product of conjugates in x²
Which expression is equivalent to (x² + 1)(x² − 1)? - Difference in area of two squares
A square tile has side length x + 4 centimetres, and a smaller square tile has side length x − 4 centimetres, where x > … - Identifying a factor of a difference of squares
Which of the following is a factor of 4x² − 25? - Reading the maximum from vertex form
The function g is defined by g(x) = −2(x − 3)² + 8. What is the maximum value of g? - Exponential growth after two periods
A population of bacteria grows by 20% every hour. If there are 500 bacteria now, how many will there be in 2 hours? - Solving a rational equation
If 12/(x − 4) = 3 and x ≠ 4, what is the value of x? - Sum of coefficients after expanding
When the expression (3x − 2)(x + 4) is written in the form ax² + bx + c, where a, b, and c are constants, what is the va… - Leading coefficient from the vertex and a point
A parabola in the xy-plane has its vertex at (2, 5) and passes through the point (3, 2). If its equation is written as y… - Sum of the roots of a shifted square
What is the sum of the solutions to (x − 5)² = 4? - Solving a rational equation with two fractions
If 1/x + 1/(2x) = 3/4 and x ≠ 0, what is the value of x? - Maximum of a quadratic in standard form
The function q is defined by q(x) = −x² + 4x + 5. What is the maximum value of q(x)? - Function from its zeros and y-intercept
A quadratic function has zeros at x = 2 and x = −3, and its graph crosses the y-axis at (0, −6). Which equation defines … - Consecutive integers with a given product
The product of two consecutive positive integers is 30. What is the sum of the two integers? - y-coordinate of a particular intersection point
In the xy-plane, the graphs of y = x² − 3 and y = 2x intersect at two points. What is the y-coordinate of the intersecti… - Constant after expanding a scaled difference of squares
When 2(x + 3)(x − 3) is written in the form ax² + c, where a and c are constants, what is the value of c? - Halving repeatedly
A cup of tea cools so that the difference between its temperature and the room's temperature halves every 10 minutes. Th… - Positive number from a quadratic word problem
A positive number is such that its square plus the number itself equals 20. What is the number? - Compound growth over two years
An investment of $5,000 grows by 5% each year. What is its value, in dollars, after 2 years? - Dimensions of a rectangle from area and a relation
A rectangle's length is 3 metres more than its width, and its area is 40 square metres. What is the width, in metres? - Number of people from a shared cost
A $12 pizza is shared equally among x people. If each person's share, reduced by $1, is $3, how many people shared the p… - Recognising a difference of squares product
A square tile has side length 3x + 1 units, and a rectangular tile has sides of 3x + 1 and 3x − 1 units. Which expressio… - Interpreting the base of an exponential model
The number of bacteria in a dish t hours after an experiment begins is modelled by N(t) = 1000(1.5)ᵗ. Which of the follo… - Positive number from a square-minus-itself relation
The square of a positive number exceeds the number by 12. What is the number? - Rational equation from a sharing context
A prize of $60 is shared equally among n winners. If there had been 2 more winners, each share would have been $5 less. … - Completing a perfect square
The expression x² + 8x + c is equal to (x + k)² for all x, where c and k are constants. What is the value of c? - Simplifying a power of a product times a power
Which expression is equivalent to (2x)³ · x²? - Identifying the function from a description of growth
A quantity starts at 80 and decreases by 25% each year. Which function gives the quantity Q after t years? - Two numbers from a sum and a product
Two positive numbers have a sum of 12 and a product of 35. What is the larger number? - A difference that is constant
For every value of x, the expression (x + 2)(x + 3) − x(x + 5) is equal to the same constant. What is that constant? - Factoring by grouping
Which expression is equivalent to 2x(x − 3) + 5(x − 3)? - Choosing a decay model
A cup of coffee is 40 degrees above room temperature, and the difference decreases by 15% every 5 minutes. Which functio… - Positive solution of a quadratic with a = 2
What is the positive solution to 2x² + 3x − 27 = 0? - Coefficient of the middle term
For all x, (x + 5)² − (x² + 25) = kx, where k is a constant. What is the value of k? - Compound growth over three periods
A town's population is 2,000 and grows by 10% each year. What will the population be after 3 years? - Positive root from a word frame
A positive number is such that its square is 35 more than twice the number. What is the number? - Product of the outer coefficients
When (3x − 2)(x + 4) is written as ax² + bx + c, what is the value of a · c? - Depreciation over two years
A laptop bought for $3,000 loses 5% of its value each year. What is its value, in dollars, after 2 years? - Two numbers from a product and a difference
Two positive numbers have a product of 40 and differ by 3. What is the smaller number? - Simplifying a perfect-square trinomial over a linear factor
For x ≠ 3, (x² − 6x + 9)/(x − 3) is equivalent to x − k. What is the value of k? - Coefficient after squaring a monomial
For all x, (3x)² − 3x² = kx², where k is a constant. What is the value of k? - Identifying the model from a percentage change
A forest's area is 8,000 hectares and shrinks by 3% each year. Which function gives the area A, in hectares, after t yea… - Two numbers, one four times the other, from a product
One positive number is four times another, and their product is 36. What is the difference between the two numbers? - Value after simplifying a rational expression
For x ≠ −5, the expression (x² + 7x + 10)/(x + 5) simplifies to x + k. What is the value of k? - Constant in a grouped factorisation
For all x, x(x + 3) − 2(x + 3) = (x + 3)(x − k). What is the value of k? - Value after two decreases of 25%
A car's value is $1,600 now and falls by 25% each year. What will its value be, in dollars, in 2 years? - Consecutive odd integers from a product
The product of two consecutive odd positive integers is 143. What is their sum? - Constant remaining after cancellation
For every value of x, the expression (x + 2)(x + 5) − x(x + 7) has the same value. What is that value? - Two increases of 20%
A plant's height increases by 20% each month. It is 30 centimetres tall now. How tall, in centimetres, will it be in 2 m… - Reading a doubling function from a table
The table gives values of a function g. Which equation could define g? - Consecutive integers from a product
The product of two consecutive positive integers is 72. What is their sum? - Factoring by grouping with a shared binomial
Which expression is equivalent to 5x(x + 2) − 3(x + 2)? - Growth by 50% over two periods
A sapling is 40 centimetres tall. Its height increases by 50% each year. How tall, in centimetres, will it be in 2 years… - Facing page numbers from their product
A book lies open at two facing pages whose page numbers have a product of 42. What is the sum of the two page numbers? - Constant in a grouped factorisation
For all x, 4x(x − 1) + 3(x − 1) = (x − 1)(4x + k). What is the value of k? - Growth of 20% over two periods
A social club has 250 members. Its membership grows by 20% each year. How many members will it have in 2 years? - Product of two consecutive odd integers from their mean
The mean of two consecutive odd integers is 10. What is their product? - Value of a scaled expression
If 2a − b = 7, what is the value of 6a − 3b + 4? - Decay of 20% over two periods
A pond holds 500 litres. Each week 20% of the water evaporates and none is added. How many litres remain after 2 weeks? - Consecutive even integers from a product
The product of two consecutive even positive integers is 80. What is their sum? - Fractional exponent of a monomial
For x > 0, which expression is equivalent to (16x⁸)^(1/4)? - Identifying a factor of a trinomial
Which of the following is a factor of 3x² + 10x − 8? - Other root of a quadratic given one root
One solution of the equation x² + kx − 15 = 0 is x = 3, where k is a constant. What is the other solution? - Sum of the numbers satisfying a quadratic condition
A number is 6 less than its square. What is the sum of all such numbers? - Distance between the x-intercepts of a parabola
The graph of f(x) = x² − 6x + 5 in the xy-plane has two x-intercepts. What is the distance between them? - Growth factor over a fixed step in an exponential function
The function f is defined by f(x) = 5 · 3^x. For any value of x, f(x + 2) is how many times f(x)? - Sum of the coefficients of an expanded square
When (2x + 3)² is expanded and written as ax² + bx + c, what is the value of a + b + c? - Perimeter of a square from its area as a trinomial
The area of a square is 9x² + 30x + 25 square units, where x > 0. Which expression gives the perimeter of the square? - Value of x² from a circle and a line through the origin
If x² + y² = 40 and y = 3x, what is the value of x²? - Exponential equation with a product of powers
If 5^x · 5^(x + 1) = 5^9, what is the value of x? - y-coordinate of the vertex from factored form
The function g is defined by g(x) = (x + 3)(x − 5). In the xy-plane, what is the y-coordinate of the vertex of the graph… - y-intercept from the x-intercepts of a monic quadratic
The graph of f(x) = x² + bx + c in the xy-plane has x-intercepts at x = −2 and x = 6, where b and c are constants. What … - Location of the minimum of an absolute-value function
The function f is defined by f(x) = |2x − 6| + 5. For what value of x does f have its minimum value? - Expression for the volume of an open box folded from a square sheet
A square of side 2 centimetres is cut from each corner of a square sheet of cardboard with side length s centimetres, wh… - Distance between the points where a horizontal line meets a parabola
In the xy-plane, the line y = 6 intersects the parabola y = x² − 3 at two points. What is the distance between the two p… - Difference of two consecutive outputs of an exponential function
The function f is defined by f(t) = 2,000(1.05)^t. What is the value of f(2) − f(1)? - Square of a sum from the sum of squares and the product
If a² + b² = 34 and ab = 15, what is the value of (a + b)²? - Radius at which a circle's area equals its circumference numerically
For what positive value of r is the area of a circle with radius r numerically equal to its circumference? - Dimension of a rectangle from an area expressed in x
A rectangular rug has sides of length (x − 3) metres and (x + 5) metres and covers an area of 20 square metres. What is … - Sum of the x-intercepts of a parabola in vertex form
In the xy-plane, the graph of g(x) = −2(x − 1)² + 8 has two x-intercepts. What is the sum of their x-coordinates? - Identifying a quadratic with no x-intercepts
Which of the following equations has a graph in the xy-plane with no x-intercepts? - Relationship between exponents with different bases
If 2^(3x) = 8^y, which of the following must be true? - Constant term of an expanded product
When (x + 1)(x + 2)(x + 3)(x + 4) is expanded, what is the constant term? - Side of a square from the area of its extension
A square garden is extended by 3 metres in one direction and 5 metres in the other, forming a rectangle with an area of … - Rational equation with a common denominator
If 2/x + 3/(2x) = 7/4, what is the value of x? - Maximum of a parabola after reflection across the x-axis
In the xy-plane, the graph of y = 2(x − 3)² − 8 is reflected across the x-axis. What is the maximum value of y on the re… - Fraction remaining after several half-lives
A radioactive substance has a half-life of 8 days. What fraction of the original amount remains after 24 days? - Value of a quadratic at zero from its vertex
The function g is defined by g(x) = (x − a)² + b, where a and b are constants. The vertex of the graph of g in the xy-pl… - Extrapolating an exponential table back to zero
The table gives values of an exponential function g. x | g(x) 1 | 6 2 | 18 3 | 54 What is the value of g(0)? - Evaluating a rational expression after simplifying
For x ≠ −3/2, the expression (4x² − 9)/(2x + 3) simplifies to a linear expression. What is the value of that expression … - Side of a square from the difference of two areas
The side of one square is 3 centimetres longer than the side of another, and the areas of the two squares differ by 51 s… - Working backwards through a constant ratio
An exponential function g satisfies g(x + 1) = (1/3) · g(x) for all x, and g(2) = 6. What is the value of g(0)? - Translating an exponential graph
The graph of y = 2ˣ in the xy-plane is translated 3 units to the right and 1 unit down. Which of the following is an equ… - Parabola from its direction and x-intercepts
Which of the following could be the equation of a parabola in the xy-plane that opens downward and has x-intercepts at x… - Average rate of change of a quadratic over an interval
The function f is defined by f(x) = x² + 1. What is the average rate of change of f from x = 1 to x = 4? - Recognising a perfect-square trinomial from a linear condition
If 2x + 3y = 12, what is the value of 4x² + 12xy + 9y²? - Sum of the coefficients of an expanded cube
When (3x − 1)³ is expanded and written as a polynomial in x, what is the sum of its coefficients? - Completing a square: sum of the constants
The expression x² + 4x + c is equal to (x + a)² for all values of x, where a and c are constants. What is the value of a… - Square whose area exceeds its perimeter by a given amount
The area of a square, in square units, is 12 more than its perimeter, in units. What is the side length of the square? - Using a parabola's symmetry to find a second input
The function f is defined by f(x) = −2x² + 8x + 1. For what value of x other than 0 is f(x) equal to f(0)? - Time from a quadratic volume model
Water flows into a tank so that the volume of water in it, in litres, t minutes after the flow starts is V(t) = t² + 6t.… - Leg of a right triangle from its area
A right triangle has legs of length x and x + 1, and its area is 15 square units. What is the value of x? - Evaluating a quadratic after finding its coefficient
The function f is defined by f(x) = ax², where a is a constant. If f(3) = 36, what is the value of f(5)? - Recognising an exponential function from a table
Each table below gives values of a function at x = 0, 1, 2, and 3. Which table could represent an exponential function? - Solution set of an absolute-value inequality
The function h is defined by h(x) = |x − 5|. Which of the following describes all values of x for which h(x) ≤ 2? - Shifting an exponent by a known amount
If 2ᵃ = 5, what is the value of 2^(a + 3)? - Sum of the roots of a quadratic with a leading coefficient
What is the sum of the solutions of 2x² − 7x + 3 = 0? - Shorter leg from a leg ratio and the hypotenuse
The legs of a right triangle are in the ratio 3 : 4, and the hypotenuse has length 15. What is the length of the shorter… - Choosing the admissible root, then evaluating
If x² − 9 = 0 and x + 3 ≠ 0, what is the value of x + 5? - y-intercept after reflecting and shifting an exponential graph
The graph of y = 2ˣ in the xy-plane is reflected across the x-axis and then shifted 3 units upward. What is the y-interc… - Change in a linear function's output for a change in input
The function f is defined by f(x) = 3x − 2. Which expression is equal to f(x + h) − f(x) for all x and h? - Evaluating a squared sum of fractional powers
For x > 0, the expression (x^(1/2) + x^(−1/2))² equals x + 2 + 1/x. What is the value of the expression when x = 4? - Sum of the solutions of a perfect-square equation
If x² − 4x + 4 = 25, what is the sum of the solutions? - Time for two cars on perpendicular roads to be a given distance apart
Two cars leave the same point at the same time, one travelling north at 30 kilometres per hour and the other east at 40 … - Symmetry of an absolute-value function
The function g is defined by g(x) = |x − a|, where a is a constant. If g(2) = g(10), what is the value of a? - Repeated halving to reach a target
A cup of coffee cools so that its temperature above room temperature halves every 8 minutes. If the coffee starts 64°C a… - Identifying a function that is decreasing everywhere
Which of the following functions is decreasing for all real values of x? - Sum of values of an odd function at symmetric inputs
The function f is defined by f(x) = x³ − x. What is the value of f(−2) + f(−1) + f(0) + f(1) + f(2)? - Combining powers with a common base
If 10ᵃ = 2 and 10ᵇ = 3, what is the value of 10^(a + b + 1)? - Difference of the squares of a sum and a difference
If ab = 6, what is the value of (a + b)² − (a − b)²? - Nonlinear system solved by substitution
If x² + y = 7 and x + y = 5, and x > 0, what is the value of x? - Value after several years of percent decline
A population of 5,000 declines by 10% each year. What is the population after 3 years? - Interval where a factored quadratic is negative
The function f is defined by f(x) = (x − 1)(x − 5). For which values of x is f(x) less than 0? - Square root of a perfect-square trinomial
For x > 0, which expression is equal to √(x² + 2x + 1)? - Coefficient and exponent of a product of monomials
The expression (3x)² · (2x)³ is equal to kxⁿ, where k and n are constants. What is the value of k + n? - Product of conjugate surds
What is the value of (2 + √3)(2 − √3)? - Evaluating a related expression after solving
If x² − 2x = 15 and x > 0, what is the value of x² + 2x? - Second input from two points on a parabola through the origin
In the xy-plane, the graph of y = ax² passes through the points (2, 12) and (k, 27), where a and k are positive constant… - Identifying a function type from second differences
The points (0, 5), (1, 8), (2, 13), and (3, 20) all lie on the graph of a function f. Which type of function could f be? - Sum of the vertex coordinates of a parabola
In the xy-plane, the graph of y = −x² + 6x has its vertex at the point (h, k). What is the value of h + k? - Composition of a quadratic and a square-root function
The functions f and g are defined by f(x) = x² − 4 and g(x) = √(x + 4). What is the value of g(f(3))? - Power of a power with a known value
If 3ˣ = 7, what is the value of 9ˣ? - Positive solution of a cubic with a common factor
If x³ = 4x and x > 0, what is the value of x? - Rational equation solved by cross-multiplying
If (x + 1)/(x − 1) = 3, what is the value of x? - Time when a projectile reaches the ground
The height, in metres, of an object t seconds after it is launched is given by h(t) = −5t² + 20t + 60. After how many se… - Exponent sum from a product of function values
The function f is defined by f(x) = 2ˣ. If f(a) · f(b) = 32, what is the value of a + b? - Constant in a product that telescopes to a difference of powers
The product (x − 1)(x + 1)(x² + 1) is equal to x⁴ − c for all x, where c is a constant. What is the value of c? - Fractional power from a known square root
If x^(1/2) = 3, what is the value of x^(3/2)? - Positive input where a quadratic equals a linear expression
For what positive value of x is x² − 2x − 3 equal to x + 1? - Time when a projectile returns to its launch height
The height, in metres, of a ball t seconds after it is thrown is h(t) = −5t² + 10t + 15. For what value of t > 0 is the … - Comparing function values at a large input
Which of the following functions has the greatest value at x = 10? - Vertex of a parabola after reflection across the y-axis
In the xy-plane, the graph of y = (x − 3)² is reflected across the y-axis. What is the x-coordinate of the vertex of the… - Numeric product that telescopes to a difference of fourth powers
What is the value of (3 + 1)(3 − 1)(3² + 1²)? - Hypotenuse from a right triangle's area and one leg
A right triangle has an area of 30 square units, and one of its legs has length 5 units. What is the length of the hypot… - Positive solution of an exponential equation with a squared exponent
If 2^(x²) = 16, what is the positive value of x? - Years for a doubling population to reach a target
A population of 400 doubles every 8 years. After how many years will the population be 3,200? - Sum of the squares of a sum and a difference
For all real a and b, the expression (a + b)² + (a − b)² is equal to k(a² + b²), where k is a constant. What is the valu… - Shifting an exponent down after solving
If 5^(x + 1) = 125, what is the value of 5^(x − 1)? - Binomial whose square is a given trinomial
The expression 4x² − 20x + 25 is the square of which of the following binomials? - Width of a rectangle from its area and a length rule
A rectangle's length is 3 units more than twice its width, and its area is 44 square units. What is the width of the rec… - Sum of a quadratic's values at symmetric integer inputs
The function g is defined by g(x) = x² − 4x + 4. What is the value of g(0) + g(1) + g(2) + g(3) + g(4)? - Height after repeated bounces
A ball dropped from a height of 8 metres rebounds to three quarters of its previous height after each bounce. What heigh… - Value of a ratio of squares under a proportional relation
If y = 3x and x ≠ 0, what is the value of (x + y)²/x²? - Coefficient of the cross term in a squared binomial
When (2x + 3y)² is expanded, what is the coefficient of the xy term? - Product from a sum and a sum of squares
If x + y = 7 and x² + y² = 29, what is the value of xy? - Positive number from the sum of itself and its square
The sum of a positive number and its square is 42. What is the number? - Volume of a cube from its surface area
A cube has a surface area of 96 square units. What is its volume, in cubic units? - Number of x-intercepts of a quartic
In the xy-plane, how many x-intercepts does the graph of y = x⁴ − 16 have? - Using symmetry about a parabola's vertex
In the xy-plane, the graph of y = a(x − 2)² + 3 passes through the point (0, 11), where a is a constant. What is the y-c… - Percent increase over two periods of compound growth
The function h is defined by h(t) = 50(1.2)ᵗ. By what percent does h increase from t = 0 to t = 2? - Using a symmetry relation of a quadratic
A quadratic function f satisfies f(x) = f(6 − x) for all values of x, and f(0) = 5. What is the value of f(6)? - Exponents from a product and a difference
If 2ᵃ · 2ᵇ = 2⁷ and a − b = 1, what is the value of a? - Constant term of an expanded square with a reciprocal
When (x + 1/x)² is expanded for x ≠ 0, what is its constant term? - Simplifying a numeric expression with powers of 2
What is the value of (2¹⁰ − 2⁹)/2⁸? - Radical equation containing a square
If √(x² + 9) = 5, what is the positive value of x? - Sum of the values of a linked variable
If x² − 5x + 6 = 0 and y = x + 1, what is the sum of all possible values of y? - Next value after finding an input from an output
The function g is defined by g(x) = 2x². If g(a) = 50 and a > 0, what is the value of g(a + 1)? - Function with a constant increase per unit step
For which of the following functions is f(x + 1) = f(x) + 3 for every value of x? - Rewriting an exponential model with a different base
The function P is defined by P(t) = 500 · 4^(t/2). Which of the following is equal to P(t) for all t? - Value of a perfect-square trinomial with a linear term
If x + y = 4, what is the value of x² + 2xy + y² − 3(x + y)? - Difference of squared binomials divided by x
For x ≠ 0, what is the value of [(x + 2)² − (x − 2)²]/x? - Leg of a right triangle with legs in ratio 1 : 2
A right triangle has legs of length x and 2x, and its hypotenuse has length √45. What is the value of x? - Circumference of a circle through a given point
In the xy-plane, a circle centred at the origin passes through the point (6, 8). The circumference of the circle can be … - Common value at two points symmetric about a vertex
In the xy-plane, the graph of f(x) = (x − 4)² + 1 passes through the points (1, k) and (7, k), where k is a constant. Wh… - Where a horizontal line meets a parabola away from the y-axis
In the xy-plane, the line y = 3 intersects the graph of y = x² − 4x + 3 at two points. What is the x-coordinate of the i… - Sum of all intercepts of a parabola
In the xy-plane, the graph of h(x) = x² − 9 has two x-intercepts and one y-intercept. What is the sum of the x-coordinat… - Numeric product that equals a difference of fourth powers
What is the value of (10² − 9²)(10² + 9²)? - Constant after dividing a difference of fourth powers
For all real x, the expression (x⁴ − 81)/(x² + 9) is equal to x² − c, where c is a constant. What is the value of c? - Power of a product from powers of its factors
If 2ˣ = 5 and 3ˣ = 7, what is the value of 6ˣ? - Perimeter of a right triangle from its hypotenuse and one leg
A right triangle has a hypotenuse of length 17 units and one leg of length 8 units. What is the perimeter of the triangl… - Cancelling a common factor with a restriction
If (x + 1)² = 4(x + 1) and x ≠ −1, what is the value of x? - Constant in a factored difference of squares
For all values of x, 3x² − 12 = 3(x − a)(x + a), where a is a positive constant. What is the value of a? - Difference of two squared binomials
For all values of x and y, (x + y)² − (x − y)² = kxy, where k is a constant. What is the value of k? - Eliminating a square between two equations
If x² + y = 10 and y = x² − 8, what is the value of y? - Rational equation with a common denominator
What value of x satisfies 3/x + 3/(2x) = 9/4? - Distance between a parabola's x-intercepts
The function f is defined by f(x) = x² − 2x − 15. The graph of f crosses the x-axis at two points. What is the distance … - Highest point of a downward parabola
In the xy-plane, the graph of y = −2(x + 1)² + 7 has its highest point at (a, b). What is the value of a + b? - Constants in a factored quadratic
For all values of x, x² + 5x − 6 = (x + a)(x − b), where a and b are positive constants. What is the value of a + b? - Constant inside a factored binomial
For all values of x, 6x² − 9x = 3x(2x − k), where k is a constant. What is the value of k? - Distance between two solutions
The equation 2x² = 32 has two solutions. What is the positive difference between them? - Side of a square from an area relation
The area of a square is equal to 4 times its side length plus 12. What is the side length of the square? - Drop in height from a quadratic model
The height of a water jet above the nozzle, in metres, x metres horizontally from the nozzle, is given by h(x) = 4 − 0.5… - Graph of a shifted exponential
In the xy-plane, the graph of y = 2ˣ − 1 passes through which of the following points? - Negative solution of a quadratic given the positive one
The equation x² + x = 30 has two solutions, one of which is 5. What is the other solution? - Perimeter of a field from its area and shape
A rectangular field is 3 times as long as it is wide, and its area is 300 square metres. What is the perimeter of the fi… - Positive solution of a quadratic with a leading coefficient
For what positive value of x is 3x² − 5x equal to 28? - Base of a power from an equation
If 3 · bˣ = 24 when x = 3, and b is positive, what is the value of b? - Coefficient of x in a product of two binomials
When (x + 5)(x − 2) is written in the form x² + bx + c, what is the value of b? - Coefficient after expanding two products
For all values of x, 2x(x − 3) − x(x + 1) = x² + kx, where k is a constant. What is the value of k? - Positive root of a quadratic in words
The square of a positive number, added to five times the number, equals 24. What is the number? - Larger solution of a shifted square
The equation (x − 4)² = 25 has two solutions. What is the larger one? - Value of a quadratic model at a point
The area A, in square centimetres, of a stain t minutes after it forms is modelled by A = 5t² − 4. What is the area afte… - Axis of symmetry from two zeros
The graph of the function f in the xy-plane crosses the x-axis at (−6, 0) and (2, 0), and f is a quadratic function. Wha… - Expression that reduces to a single power
For all values of x, (x + 6)(x − 6) + 36 is equivalent to which of the following? - Sum of coefficients after simplifying
When 5x(x + 2) − 3x² is written as ax² + bx, what is the value of a + b? - Number from a product of a sum and a difference
The product of 2 more than a number and 2 less than the same number is 96. If the number is positive, what is it? - Solution of a rational equation
What value of x satisfies 2/x + 1/x = 1? - Input from an output of a quadratic function
The function q is defined by q(x) = x² − 5x. For what positive value of x is q(x) = 6? - Two triplings of a population
A population of 800 bacteria triples every 4 hours. What is the population after 8 hours? - Number from a square and triple that are equal apart
The square of a positive number exceeds three times the number by 18. What is the number? - Three doublings from 60
A quantity of 60 doubles every 3 hours. What is the quantity after 9 hours? - Constant after expanding and combining two products
For all values of x, x(x + 3) − 3(x + 3) = x² − k, where k is a constant. What is the value of k? - Three doublings from 40
A quantity starts at 40 and doubles every 6 hours. What is the quantity after 18 hours? - Smaller of two numbers from a difference and a product
Two positive numbers differ by 5, and their product is 84. What is the smaller of the two numbers? - Solving a line-parabola system
If (x, y) is a solution to the system y = x² and y = 4x − 3, and x > 2, what is the value of y? - A number plus its reciprocal
A number greater than 1 added to its reciprocal equals 5/2. What is the number? - Difference of two squared binomials
If (x + 4)² − (x − 4)² = kx for all values of x, what is the value of k? - Squaring a monomial with two variables
Which of the following is equivalent to (3x²y)²? - Dividing monomials with two variables
For x ≠ 0 and y ≠ 0, which expression is equivalent to 6x²y³ / (2xy)? - Constant after completing the square
The expression x² − 10x + 34 can be written in the form (x − h)² + k, where h and k are constants. What is the value of … - Perimeter of a rectangle from its area
A rectangle's length is 3 times its width, and its area is 108 square units. What is the perimeter of the rectangle? - Smallest positive solution of a quartic
What is the smallest positive solution to the equation x⁴ = 16x²? - Quotient of a difference of cubes
For all x ≠ 2, (x³ − 8)/(x − 2) = x² + ax + b, where a and b are constants. What is the value of a + b? - Dividing a binomial by a monomial
For x ≠ 0, which expression is equivalent to (2x² + 6x)/(2x)? - Product of the solutions of a quadratic
What is the product of the two solutions to x² + 2x − 8 = 0? - Expanding a scaled squared binomial
Which expression is equivalent to 2(x − 3)² + 5? - Original side of a square from an area increase
When the side of a square is increased by 3, its area increases by 39. What was the original side length? - Sum of the coefficients of a polynomial sum
The sum of the expressions 3x² − 2x + 1 and −x² + 5x − 4 can be written as ax² + bx + c, where a, b, and c are constants… - Larger of two numbers from a ratio and a product
Two positive numbers are in the ratio 2 to 3, and their product is 96. What is the larger number? - Sum of two simplified fractions
For x ≠ 0, which expression is equivalent to (x² + 3x)/x + (2x − 4)/2? - Positive solution of a proportion with x on both sides
For what positive value of x is x/4 equal to 9/x? - X-intercepts of a quadratic function
The function f is defined by f(x) = x² − 6x + 5. At which values of x does the graph of f cross the x-axis? - Yearly growth rate from an exponential model
The population of a town t years from now is modelled by P(t) = 12,000(1.03)ᵗ. By what percent does the model say the po… - Range of a shifted square function
The function f is defined by f(x) = (x − 2)² + 3. Which of the following describes all possible values of f(x)? - Parabola with a given intercept that opens downward
Which of the following functions has a graph in the xy-plane that opens downward and has a y-intercept of 5? - Factoring out a common binomial
Which expression is equivalent to 2x(x + 3) + (x + 3)? - Positive value from a substitution
If x² = 2y and y = 8, what is the positive value of x? - x² coefficient in a binomial-trinomial product
When (x + 3)(x² − 2x + 4) is expanded, what is the coefficient of x²? - What remains after removing the x² part
For every value of x, (2x − 5)(3x + 4) − 6x² = bx + c. What is the value of b? - Difference of squares with leading coefficient
The expression 4x² − 81 factors as (2x − a)(2x + a) for a positive constant a. What is the value of a? - Quadratic left after cancelling cubes
For every value of x, the difference (x + 2)³ − x³ equals ax² + bx + c. What is the value of a? - Adding fractions with x denominators
For x ≠ 0, the sum 2/x + 3/(2x) is equivalent to a/(2x) for a constant a. What is the value of a? - Cube root of a monomial
For positive x and y, which expression is equivalent to the cube root of x⁶y³? - Exponential equation solved by collecting terms
If 2ˣ⁺⁴ = 5 · 2ˣ + 88, what is the value of x? - Zeros of a biquadratic function
The function f(x) = x⁴ − 20x² + 64 has four real zeros. What is the sum of the two largest zeros? - Where a line meets a parabola
In the xy-plane, the line y = 2x and the parabola y = x² intersect at two points. What is the sum of the x-coordinates o… - Gap between a quadratic's roots
The solutions of x² − 11x + 28 = 0 are p and q, with p > q. What is the value of p − q? - Radical equation hiding a square
If √(x² + 9) = 5 and x > 0, what is the value of x? - Rational equation in one step
If 6/x + 2 = 5 and x ≠ 0, what is the value of x? - Minimum from scaled vertex form
What is the minimum value of the function f(x) = 2(x − 3)² − 5? - Evaluating a factored quadratic at a negative input
The function p is defined by p(x) = (x − 5)(x + 2). What is the value of p(−3)? - x-intercept of a shifted exponential
In the xy-plane, at what value of x does the graph of y = 3ˣ − 9 cross the x-axis? - Maximum of a downward parabola
What is the maximum value of the function f(x) = −(x + 1)² + 8?
Hard (314)
- Exponential functions
A population of bacteria doubles every 3 hours. If it starts at 500, which function models the population P after t hour… - Discriminant and real solutions
How many real solutions does x² + 4x + 5 = 0 have? - Exponential decay
A car worth $20,000 loses 15% of its value each year. What is its value after 2 years? - Exponential growth modeling
An investment of $1,000 grows at 6% per year, compounded annually. Which expression gives its value after t years? - Composition of functions
If f(x) = x² − 1 and g(x) = x + 2, what is g(f(3))? - Vertex and axis of symmetry
The parabola y = x² − 8x + 12 crosses the x-axis at two points. What is the x-coordinate of the vertex? - Radical equations and extraneous solutions
What is the solution set of the equation x − √(x + 5) = 1 ? - Solving exponential equations by rewriting with a common base
If (1/8)^x · 4^(2x + 1) = 16^(x − 3), what is the value of x ? - Simplifying a rational expression
For x ≠ 2, which expression is equivalent to (x² − 4) / (x − 2)? - Raising a product to a power
Which expression is equivalent to (3x²y)³? - Evaluating an exponential decay model
According to the model, what is the value of the equipment, in dollars, after 2 years? - Maximum value of a downward-opening parabola
The function f is defined by f(x) = −2(x − 1)² + 7. What is the maximum value of f(x)? - Using the discriminant
How many real solutions does the equation 2x² + 3x + 5 = 0 have? - Radical equation with an extraneous root
What is the solution to the equation √(x + 7) = x − 5? - Simplifying a rational expression by factoring
For x ≠ −3, which expression is equivalent to (2x² + 6x) / (x + 3)? - Square root of a monomial
For x > 0, which expression is equivalent to √(9x⁴)? - Rewriting an exponential with a different base
Which expression is equivalent to 8ˣ? - Comparing linear and exponential growth
The functions f and g are defined by f(x) = 3x + 10 and g(x) = 2ˣ. What is the value of g(6) − f(6)? - Vertex from standard form
In the xy-plane, what are the coordinates of the vertex of the graph of y = x² + 4x − 1? - Rational equation
If 6/x = x − 1 and x ≠ 0, what is the sum of all solutions to the equation? - Exponential equation with matching bases
If 2ˣ⁺¹ = 32, what is the value of x? - Vertical shift of a parabola
The function f is defined by f(x) = x² − 4, and the function g is defined by g(x) = f(x) + 3. What are the coordinates o… - Counting intersections of a line and a parabola
In the xy-plane, how many points do the graphs of y = x² + 1 and y = x have in common? - Solving a radical equation
What value of x satisfies √(2x + 3) = 5? - Multiplying rational expressions
For x ≠ −1 and x ≠ 2, which expression is equivalent to (x / (x + 1)) · ((x + 1) / (x − 2))? - Binomial times a trinomial
Which expression is equivalent to (2x + 3)(x² − x + 1)? - Power of a negative fractional exponent
For x > 0, which expression is equivalent to (x^(−1/2))⁴? - Translating a parabola
The graph of the function g in the xy-plane is the graph of f(x) = x² shifted 3 units to the right and 2 units down. Whi… - Writing a half-life model
A patient takes a 60-milligram dose of a medicine. The amount of the medicine in the bloodstream halves every 4 hours. W… - Product of intersection x-coordinates
In the xy-plane, the graphs of y = x² − 3x and y = 4 intersect at two points. What is the product of the x-coordinates o… - Negative exponent on a fraction
For x ≠ 0 and y ≠ 0, which expression is equivalent to (2x/y)⁻²? - Division with a remainder
For x ≠ −2, which expression is equivalent to (x² + 5x + 7)/(x + 2)? - Composing a function with itself
The function f is defined by f(x) = x² + 1. What is the value of f(f(2))? - Quadratic from a rectangle's area
A rectangular garden has an area of 60 square metres, and its length is 7 metres more than its width. What is the width … - Exponential equation with a common base
For what value of x is 9ˣ equal to 27? - Difference of two squared binomials
Which expression is equivalent to (x + y)² − (x − y)²? - Squaring a difference with a reciprocal term
For x ≠ 0, which expression is equivalent to (2x − 3/x)²? - Factoring by grouping, then a difference of squares
Which expression is equivalent to x³ − 3x² − 4x + 12? - Combining a product and a square
Which expression is equivalent to (2x − 1)(2x + 1) − (x − 3)²? - Finding the leading coefficient from a point
The function f is defined by f(x) = a(x − 2)² − 8, where a is a constant. If f(0) = 0, what is the value of f(5)? - Rational equation that becomes a quadratic
If x + 6/x = 5 and x ≠ 0, what is the product of all the solutions to the equation? - Splitting an exponent
Which expression is equivalent to 4ˣ⁺¹? - Maximum of an absolute-value function
The function h is defined by h(x) = 5 − |x − 1|. What is the maximum value of h(x)? - Rational equation with two roots
If 4/(x − 3) = x and x ≠ 3, which of the following gives all solutions to the equation? - Squared binomial minus a square term
Which expression is equivalent to (3x − 2)² − 9x²? - Inputs that give a stated output
The function f is defined by f(x) = 3(x − 1)² + 2. For what values of x does f(x) = 14, and what is their sum? - Real solutions of a quadratic in x²
How many real solutions does the equation x⁴ − 5x² + 4 = 0 have? - Base of an exponential through two points
The graph of the exponential function h(x) = a · bˣ, where a and b are positive constants, passes through the points (0,… - Consecutive integers from a sum of squares
The sum of the squares of two consecutive positive integers is 61. What is the larger of the two integers? - Sum of all real solutions of a cubic
If x³ = 9x, what is the sum of all real solutions to the equation? - Power of a fractional power
For x > 0, which expression is equivalent to (x^(2/3))^(3/4)? - Quadratic with no x term from two values
The function f is defined by f(x) = ax² + c, where a and c are constants. If f(0) = 3 and f(2) = 11, what is the value o… - Consecutive even integers from a product
The product of two consecutive even integers is 48. If both integers are positive, what is the smaller one? - Squaring a binomial with a square root
For x ≥ 0, which expression is equivalent to (√x + 2)²? - Sum of the x-intercepts from vertex form
The function f is defined by f(x) = −2(x − 3)² + 8. What is the sum of the x-intercepts of the graph of f in the xy-plan… - Second intersection of a parabola and a line
In the xy-plane, the parabola y = x² + 1 and the line y = 2x + 1 both pass through the point (0, 1). What is the x-coord… - Squaring a binomial with a fraction
Which expression is equivalent to (x + 1/2)²? - Starting value of a tripling quantity
A quantity triples every hour. Two hours after the start, the quantity is 18. What was the quantity at the start? - Proportion that becomes a quadratic
If 1/x = x/4 and x > 0, what is the value of x? - Product minus a neighbouring square
Which expression is equivalent to x(x + 2) − (x + 1)²? - Time for a drug to fall to a level
A patient has 400 milligrams of a drug in the bloodstream, and the amount halves every 3 hours. After how many hours wil… - Smaller intersection of a line and a parabola
In the xy-plane, the line y = 5x − 6 meets the parabola y = x² at two points. What is the smaller of the two x-coordinat… - Squared difference plus twice the product
Which expression is equivalent to (a − b)² + 2ab? - First power to exceed a threshold
For which positive integer n is 3ⁿ the first power of 3 that is greater than 200? - Sum of roots after rearranging a product
If x(x − 3) = 4, what is the sum of all the solutions to the equation? - Simplifying a squared monomial over a monomial
For x ≠ 0 and y ≠ 0, which expression is equivalent to (x³y)² / (xy²)? - Coefficient in a product of binomials
If (2x + a)(x − 5) = 2x² − 7x − 15 for all values of x, what is the value of a? - Axis of symmetry from the x-intercepts
The graph of f(x) = x² − 6x + 5 in the xy-plane crosses the x-axis at two points. What is the equation of the graph's ax… - When a projectile returns to the ground
The height of a ball t seconds after being thrown upward is given by h(t) = 40t − 5t². After how many seconds does the b… - Sum of the solutions of a quadratic
What is the sum of all solutions to the equation x² + x − 6 = 0? - Solutions of a linear-quadratic system
Which of the following ordered pairs (x, y) is a solution to the system below? y = x² − 1 y = x + 1 - Constant in a simplified rational expression
For x ≠ 0 and x ≠ −3, the expression (x² − 9)/(x² + 3x) is equivalent to (x − a)/x, where a is a constant. What is the v… - Vertex form to x-intercepts
The function h is defined by h(x) = (x − 4)² − 9. At which values of x does the graph of h cross the x-axis? - Exponential decay after three periods
A car's value decreases by 10% each year. If the car is worth $800 now, what will it be worth, in dollars, in 3 years? - Number of solutions of a tangent system
How many solutions does the system y = x² and y = 6x − 9 have? - Larger root of a quadratic with leading coefficient 2
What is the greater of the two solutions to 2x² − 5x − 12 = 0? - Constant in a simplified difference
For all values of x, (x + 3)² − (x − 1)(x + 1) = ax + b, where a and b are constants. What is the value of a + b? - Choosing the growth model from a description
A savings account starts with $500 and earns 3% interest, compounded once a year, with no other deposits. Which function… - Two numbers from a sum and a sum of squares
Two positive numbers have a sum of 7 and a sum of squares of 25. What is the product of the two numbers? - Radical equation solved by squaring
What is the solution to √(3x + 1) = x − 1? - Maximum height and when it occurs
The height, in feet, of a ball t seconds after it is thrown is given by h(t) = −16t² + 64t. What is the maximum height t… - Minimum value from factored form
The function k is defined by k(x) = (x − 2)(x − 6). What is the minimum value of k(x)? - Perfect-square trinomial constant
For what value of c is x² + 10x + c a perfect square of a binomial? - Sum of the solutions of a quadratic with a ≠ 1
What is the sum of the solutions to 2x² − x − 15 = 0? - Constant in a factored form
For all x, 3x(x − 4) + 2(x − 4) = (3x + k)(x − 4), where k is a constant. What is the value of k? - Vertex and intercepts from vertex form
The function w is defined by w(x) = (x + 1)² − 4. Which statement about the graph of y = w(x) is true? - Comparing growth over equal intervals
The function g is defined by g(x) = 2ˣ. Which statement compares the increase in g from x = 1 to x = 3 with the increase… - Both roots of a quadratic with no linear term, in a word frame
Four times the square of a positive number is 49. What is the number? - Two numbers from their product and difference
Two positive numbers differ by 2 and have a product of 24. What is their sum? - Coefficient after simplifying a difference
For all x, (x − 5)(x + 5) − (x − 5)² = ax + b, where a and b are constants. What is the value of a? - Constant in a simplified rational expression
For x ≠ 0 and x ≠ 4, (x² − 16)/(x² − 4x) is equivalent to (x + k)/x, where k is a constant. What is the value of k? - Range of a downward parabola
The function p is defined by p(x) = −(x − 3)² + 7. Which of the following describes all possible values of p(x)? - Distance between two intersection points
In the xy-plane, the graphs of y = x² − 2x and y = 3 intersect at two points. What is the distance between these two poi… - Value of a simplified rational expression
For x ≠ 2 and x ≠ −2, the expression (x² + 2x)/(x² − 4) simplifies to x/(x − k), where k is a constant. What is the valu… - Time at which a projectile is highest
The height of a rocket t seconds after launch is modelled by h(t) = −5t² + 30t. At what time is the rocket at its maximu… - Sum of squares from sum and product
Two numbers have a sum of 6 and a product of 8. What is the sum of their squares? - Intersection of a circle and a line through the origin
In the xy-plane, the circle x² + y² = 20 and the line y = 2x intersect at two points. Which of the following is one of t… - Constant from cancelling a difference of squares
For x ≠ −3/2, (4x² − 9)/(2x + 3) is equivalent to 2x − c, where c is a constant. What is the value of c? - Minimum value from vertex form with a coefficient
The function d is defined by d(x) = 2(x + 1)² − 8. Which statement is true? - System with a quadratic: number of solutions
How many solutions does the system y = x² + 1 and y = 1 − x² have? - Two numbers, one triple the other, from a product
One positive number is three times another, and their product is 27. What is the sum of the two numbers? - Sum of two squared binomials
Which expression is equivalent to (x − 2)² + (x + 2)²? - Coefficient in an expanded product of three factors
When x(x + 1)(x + 2) is written as x³ + ax² + bx, what is the value of a + b? - Axis of symmetry and minimum from factored form
The function q is defined by q(x) = (x − 5)(x + 1). Which statement is true about the graph of y = q(x)? - Triangle dimensions from area and a relation
A right triangle's longer leg is 4 units more than its shorter leg, and its area is 48 square units. What is the length … - Difference of two numbers from sum and product
Two positive numbers have a sum of 13 and a product of 36. What is the positive difference between them? - Constant after simplifying a rational expression with a common factor
For x ≠ −2, (5x² − 20)/(5x + 10) is equivalent to x − k, where k is a constant. What is the value of k? - Coefficient from a difference of areas
A rectangular garden has length 2x + 5 metres and width x − 1 metres, where x > 1. A square patio of side x metres is bu… - Where a function is positive
The function m is defined by m(x) = −(x − 2)² + 9. For which values of x is m(x) > 0? - Choosing the equation that models a quadratic situation
A rectangular poster has an area of 600 square centimetres, and its height is 10 centimetres more than its width w. Whic… - Chord length from a circle and a vertical line
In the xy-plane, the circle x² + y² = 25 and the line x = 3 intersect at two points. What is the distance between these … - Subtracting fractions with the same denominator
For x ≠ −1, which expression is equivalent to x/(x + 1) − 1/(x + 1)? - Reading intercepts from vertex form
The function n is defined by n(x) = (x − 1)² − 16. Which statement about the graph of y = n(x) is true? - Legs of a right triangle from the hypotenuse and a relation
A right triangle has a hypotenuse of 10 units, and one leg is 2 units longer than the other. What is the length of the s… - Which point is a solution of a nonlinear system
Which ordered pair (x, y) satisfies BOTH y = x² − 3 and y = 2x? - Sum of coefficients after combining two products
For all x, (x − 4)(x + 4) + (x − 4)² = ax² + bx + c. What is the value of a + b + c? - Minimum and axis from vertex form with a coefficient
The function r is defined by r(x) = 3(x − 2)² + 1. Which statement is true? - Product of the roots
What is the product of the solutions to x² − 5x + 4 = 0? - Distance between the points where a horizontal line meets a parabola
In the xy-plane, the line y = 4 intersects the graph of y = x² − 5 at two points. What is the distance between those poi… - Constant in a simplified quotient
For x ≠ 0 and x ≠ −5, (x² − 25)/(x² + 5x) is equivalent to (x − k)/x, where k is a constant. What is the value of k? - Maximum and zeros from vertex form
The function u is defined by u(x) = −2(x + 1)² + 8. Which statement is true? - Triangle legs from area and a difference
The two legs of a right triangle differ by 7 centimetres, and the triangle's area is 30 square centimetres. What is the … - Longer side from perimeter and area
A rectangle has a perimeter of 30 centimetres and an area of 56 square centimetres. What is the length, in centimetres, … - Constant term after two products cancel their squares
For all x, (2x + 3)(x − 1) − 2x(x + 1) = ax + b, where a and b are constants. What is the value of b? - Ratio of two rectangles' areas
Rectangle P has sides of length x and x + 4, and rectangle Q has sides of length x − 4 and x + 4, where x > 4. The ratio… - Vertex from factored form
The function z is defined by z(x) = (x − 6)(x + 2). What are the coordinates of the vertex of the graph of y = z(x)? - Where a downward parabola is non-negative
The function g is defined by g(x) = −(x + 3)² + 4. For which values of x is g(x) ≥ 0? - Rectangle sides from area and a difference
A rectangle's length is 5 metres more than its width, and its area is 150 square metres. What is its perimeter, in metre… - Sum of coordinates of an intersection point
In the xy-plane, the circle x² + y² = 50 and the line y = x intersect at two points. What is the sum of the x-coordinate… - A difference of two products that is constant
For every value of x, the expression (x + 5)(x − 2) − (x + 1)(x + 2) has the same value. What is that value? - Value of a related expression
If 3x + 5 = 20, what is the value of 6x + 10? - Minimum value from factored form
The function v is defined by v(x) = (x + 4)(x − 2). What is the minimum value of v(x)? - Interpreting a quadratic model's intercept
The height h, in metres, of a stone t seconds after it is thrown from a bridge is modelled by h(t) = −5t² + 10t + 15. Wh… - Rectangle's diagonal from area and a relation
A rectangle's length is 3 units more than its width, and its area is 40 square units. What is the length of its diagonal… - Number of intersection points
How many points do the graphs of y = x² and y = 4x − 4 have in common? - Sum of the remaining coefficients
For all x, (3x + 1)(x + 2) − 3x(x + 3) = ax + b, where a and b are constants. What is the value of a + b? - Maximum and zeros from vertex form
The function w is defined by w(x) = −(x − 4)² + 9. Which statement is true? - Interpreting a quadratic model's constant
The height h, in metres, of a ball t seconds after it is hit is modelled by h(t) = −5t² + 12t + 1. Which statement is th… - Perimeter from area and a relation
A rectangle's length is 3 units more than its width, and its area is 54 square units. What is its perimeter, in units? - Distance between the intersection points of two parabolas
In the xy-plane, the graphs of y = x² − 4 and y = −x² + 4 intersect at two points. What is the distance between those po… - Value of x from a difference of squared binomials
For what value of x is (2x − 5)² − (2x + 5)² equal to −120? - Sum of x-coordinates where a line meets a circle
The line y = x + 1 intersects the circle x² + y² = 25 at two points. What is the sum of the x-coordinates of the two poi… - Finding a quadratic coefficient from two points
The graph of y = ax² + bx in the xy-plane passes through the points (1, 5) and (2, 14), where a and b are constants. Wha… - Time interval a projectile stays at or above its launch height
A ball is thrown upward from a platform, and its height in metres t seconds after the throw is h(t) = −5t² + 20t + 25. F… - Vertex form: sum of the parameters
The expression 2x² − 12x + 23 can be written in the form 2(x − h)² + k, where h and k are constants. What is the value o… - Consecutive even integers: product exceeds sum
The product of two positive consecutive even integers is 34 more than their sum. What is the larger of the two integers? - Two numbers from their difference and the sum of their squares
Two positive numbers differ by 5, and the sum of their squares is 97. What is the larger of the two numbers? - First year a decaying value falls below half
A machine loses 20% of its value each year. After how many whole years is its value first less than half of its original… - Combining radicals with the same radicand
For x > 0, the expression √(50x⁴) − √(8x⁴) can be written as kx²√2, where k is a constant. What is the value of k? - Constant that removes the linear term of a product
When (3x + a)(x − 2) is expanded, the result has no x term, where a is a constant. What is the value of a? - Numerator after combining two rational expressions
For x ≠ 1 and x ≠ −2, the expression 4/(x − 1) − 3/(x + 2) is written as a single fraction with denominator (x − 1)(x + … - Right triangle with the hypotenuse described in terms of a leg
In a right triangle, the shorter leg is 8 units long and the hypotenuse is 2 units longer than the longer leg. What is t… - Exponential equation with two different bases
If 4^x = 8^(x − 1), what is the value of x? - Evaluating a quadratic after finding its leading coefficient
In the xy-plane, the graph of y = a(x − 2)(x − 6) passes through the point (4, −8), where a is a constant. What is the v… - Where a quadratic exceeds its constant term
The function h is defined by h(x) = −x² + 4x + 12. For which values of x is h(x) greater than 12? - Sum of the x-intercepts of a cubic
In the xy-plane, the graph of f(x) = x³ − 4x crosses the x-axis at three points. What is the sum of the x-coordinates of… - Product of a square root and a cube root as a single power
For x > 0, the product √x · ∛x can be written as x^k, where k is a constant. What is the value of k? - Sum of the constants in a factorisation
The polynomial 6x² + 7x − 20 is equivalent to (2x + a)(3x + b) for all x, where a and b are integers. What is the value … - Constant term from the sum and difference of roots
The equation x² − 6x + c = 0 has two real solutions that differ by 4, where c is a constant. What is the value of c? - Where consecutive outputs of an exponential function differ by a given amount
The function f is defined by f(x) = 2^x. For what value of x is f(x + 1) − f(x) = 64? - Solving for the time in an exponential growth model
The population of an insect colony t weeks after observation begins is modelled by n(t) = 50 · 3^(t/2). After how many w… - Sum of the times a projectile is at a given height
The height, in metres, of a ball t seconds after it is thrown is given by h(t) = 40t − 5t². The ball is at a height of 6… - Coefficient of x² in a product of three linear factors
When (x − 2)(x + 3)(x − 5) is expanded, what is the coefficient of x²? - Linear form of a difference of a square and a monomial
The expression 3(x − 4)² − 3x² can be rewritten in the form ax + b, where a and b are constants. What is the value of a? - Sum of two numbers from their product and the sum of their squares
The product of two positive numbers is 36, and the sum of their squares is 97. What is the sum of the two numbers? - Shifting the input of an exponential function by a half
The function f is defined by f(x) = 4ˣ. Which expression is equal to f(x + 1/2) for all x? - Quadratic from its zeros and y-intercept
A quadratic function f satisfies f(1) = 0, f(7) = 0, and f(0) = 14. What is the value of f(4)? - Sum of the remaining roots after removing a known factor
The polynomial x³ − 7x + 6 has (x − 1) as a factor. What is the sum of the other two roots of the polynomial? - Exponents after simplifying a quotient of monomials
For nonzero x and y, the expression x⁻²y³ / (xy⁻¹) is equivalent to xᵃyᵇ, where a and b are constants. What is the value… - Simplifying the difference of function values
The function f is defined by f(x) = x² − 3x. Which expression is equal to f(x + 2) − f(x) for all x? - Exponential equation whose exponents form a quadratic
If 3^(x²) = 3^(5x − 6), what is the sum of all possible values of x? - Rectangle whose area is unchanged after resizing
A rectangle's length is twice its width. If the width is increased by 2 units and the length is decreased by 3 units, th… - Counting integer inputs below a threshold for a quadratic
The function f is defined by f(x) = (x − 3)² + 1. For how many integer values of x is f(x) < 10? - Product of the solutions of a squared-binomial equation
What is the product of the solutions of the equation (x − 4)² = 2x − 5? - Radius of a cylinder from its volume and a height relation
A cylinder's height is twice its radius, and its volume is 54π cubic centimetres. What is the radius of the cylinder, in… - Point on a line at a given distance from the origin
In the xy-plane, the point (x, 2x), where x > 0, is at a distance of 5√5 from the origin. What is the value of x? - Difference of two squared expressions with a reciprocal
For x ≠ 0, what is the value of (x + 1/x)² − (x − 1/x)²? - Factoring out a common binomial
The expression (2x + 1)(x − 3) − (x − 3)² can be written as (x − 3)(x + k), where k is a constant. What is the value of … - Equation with a common factor on both sides
What is the sum of all solutions of the equation (x − 2)(x + 5) = (x − 2)(2x − 1)? - Factoring a difference of consecutive powers
The function f is defined by f(x) = 3ˣ − 3^(x − 1). For every value of x, f(x) can be written as k · 3^(x − 1), where k … - Smallest integer input where an exponential function exceeds a threshold
The function g is defined by g(x) = 100(1.1)ˣ. What is the smallest integer value of x for which g(x) is greater than 13… - Counting the distinct real zeros of a polynomial
How many distinct real values of x satisfy 2x³ − 6x² + 4x = 0? - Rational equation whose only candidate is excluded
How many solutions does the equation x/(x + 1) = 3/(x + 1) + 1 have? - Counting integer inputs where an exponential lies in a range
For how many integer values of x is 2ˣ greater than 10 and less than 1,000? - Factoring a difference of squares with a binomial
The expression 16 − (x − 3)² can be written as (a − x)(x + b), where a and b are positive constants. What is the value o… - Simplifying a numeric quotient by factoring
What is the value of (10⁶ − 10⁴)/(10² − 1)? - Square root of a perfect square with a negative variable
If x < 0, which expression is equal to √(9x²)? - Product of the solutions of a rational equation
The equation 2/x = x − 1 has two solutions. What is the product of the solutions? - Translating a parabola so its vertex is at the origin
In the xy-plane, the graph of y = x² − 2x − 3 is translated so that its vertex lies at the origin. Which of the followin… - Factoring a quartic as a product of two quadratics
The expression x⁴ − 5x² + 4 is equal to (x² − a)(x² − b) for all x, where a and b are constants with a > b. What is the … - Coefficient of a mixed term in a binomial cube
When (x + y)³ is expanded, what is the coefficient of the x²y term? - Remainder from a polynomial identity
A polynomial p satisfies p(x) = (x − 2)q(x) + 5 for all x, where q is another polynomial. What is the value of p(2)? - Sum of squares equal to zero
The equation (x + 1)² + (y − 2)² = 0 has exactly one solution (x, y) in real numbers. What is the value of x + y? - Input where a doubled argument multiplies an exponential by 8
The function g is defined by g(x) = 2ˣ. For what value of x is g(2x) = 8 · g(x)? - Value at zero from a vertex and a point
A quadratic function f has its vertex at (2, −9), and its graph in the xy-plane passes through the point (5, 0). What is… - Sum from a reciprocal sum and a product
If 1/x + 1/y = 7/12 and xy = 12, what is the value of x + y? - Area of an isosceles right triangle from its hypotenuse
An isosceles right triangle has a hypotenuse of length 10. What is the area of the triangle? - Cube whose volume is a multiple of its surface area
A cube's volume, in cubic units, is numerically equal to 6 times its surface area, in square units. What is the side len… - Expanding the square of a sum of surds
The expression (√5 + √2)² can be written as a + b√10, where a and b are integers. What is the value of a + b? - Sum of the squares of two numbers from their sum and product
The sum of two numbers is 8 and their product is 13. What is the sum of the squares of the two numbers? - Sum of squares from a squared difference and a product
If (x − y)² = 9 and xy = 10, what is the value of x² + y²? - Reciprocal of a sum of reciprocals
For nonzero x and y with x + y ≠ 0, which expression is equal to (x⁻¹ + y⁻¹)⁻¹? - Width of a rectangle from its diagonal and a length ratio
A rectangle's length is twice its width, and its diagonal has length 5√5. What is the width of the rectangle? - Minimum of a sum of two absolute values
The function f is defined by f(x) = |x| + |x − 4|. What is the minimum value of f? - Coefficient of x² in the square of a trinomial
When (x² + x + 1)² is expanded, what is the coefficient of x²? - Combining powers with different exponents
If 3ᵃ = 4 and 3ᵇ = 5, what is the value of 3^(2a − b)? - Longer side of a rectangle from its area and perimeter
A rectangular field has an area of 300 square metres and a perimeter of 70 metres. What is the length, in metres, of its… - Least value of a quadratic on a restricted interval
The function f is defined by f(x) = x² − 6x + 10. What is the least value of f(x) for 0 ≤ x ≤ 2? - Minimum of a sum of two squared binomials
What is the minimum value of the function f(x) = (x − 3)² + (x + 1)²? - Product of all real solutions of a quartic in x²
The equation x⁴ − 10x² + 9 = 0 has four real solutions. What is the product of the four solutions? - Side of a square inscribed in a circle of known area
A square is inscribed in a circle whose area is 18π square units. What is the side length of the square? - Number of sides from a count of diagonals
A polygon with n sides has n(n − 3)/2 diagonals. How many sides does a polygon with 20 diagonals have? - Factoring by grouping
The expression x²(x − 1) − 4(x − 1) factors completely as (x − 1)(x + a)(x − a), where a is a positive constant. What is… - Base side of a box from a cubic volume equation
A box has a square base of side s units and a height of s + 2 units. Its volume is 45 cubic units. What is the value of … - Difference of two rational expressions that reduces to a constant
For x ≠ −1 and x ≠ 2, the expression (x² − 1)/(x + 1) − (x² − 4)/(x − 2) is equal to a constant. What is that constant? - Exponential equation with a repeated term
If 4ˣ + 4ˣ = 2⁷, what is the value of x? - Rational equation whose quadratic terms cancel
For what value of x does (x − 1)/(x + 2) equal (x + 1)/(x − 2)? - Number of terms from a triangular-number formula
The sum of the first n positive integers is given by n(n + 1)/2. For what value of n is this sum equal to 78? - Square of a sum of surds that simplifies to an integer
What is the value of (√8 + √2)²? - Sum of the squares of a quadratic's roots
The equation x² − 2x − 8 = 0 has roots p and q. What is the value of p² + q²? - Exponential equation with no solution
The functions f and g are defined by f(x) = 3 · 2ˣ and g(x) = 2ˣ. For which value of x is f(x) = g(x + 2)? - Numeric check of the cube-of-a-sum identity
If a = 3 and b = −2, what is the value of (a + b)³ − (a³ + b³)? - Value at the midpoint of two zeros
The function f is defined by f(x) = x² − 6x + 5 and has zeros at x = a and x = b. What is the value of f at the midpoint… - Ratio of two outputs of an exponential
The function g is defined by g(x) = 3·2ˣ. What is the value of g(5)/g(2)? - Input for a target output of an exponential minus one
The function p is defined by p(x) = 2ˣ − 1. For what value of x is p(x) = 31? - Population from a model with a halved exponent
A population is modelled by P(t) = 200·3^(t/2), where t is the number of years after 2020. What does the model give for … - Same zeros, opposite direction
Which of the following functions has the same zeros as f(x) = x² − 9 but a graph that opens downward? - Height of a parabolic arch off centre
A parabolic arch is 10 metres wide at ground level and 5 metres high at its centre. How high, in metres, is the arch at … - Linear quotient of a quadratic
For x ≠ −3, the expression (2x² + 7x + 3)/(x + 3) is equal to ax + b, where a and b are constants. What is the value of … - Integer from a sum of related square roots
For what positive integer n is √n + √(4n) equal to 9? - Combining a power and a square root
For x > 0, the expression x⁴ · √x can be written as xⁿ, where n is a constant. What is the value of n? - Product of the x-coordinates where a line meets a parabola
In the xy-plane, the graphs of y = x² − 6x + 8 and y = 2x − 7 meet at two points whose x-coordinates are p and q. What i… - Radical equation with an extraneous root
What is the solution to the equation √(2x + 7) = x + 2? - Halving time for a cooling cup
The amount by which a cup of coffee's temperature exceeds room temperature halves every 10 minutes. If the excess is 80°… - Leading coefficient from the y-intercept
The function f is defined by f(x) = a(x − 2)(x − 6), where a is a constant. If the graph of f passes through (0, 24), wh… - Constant term from a known minimum
The function g is defined by g(x) = x² − 6x + c, where c is a constant. If the minimum value of g is −4, what is the val… - Constant after cancelling a squared factor
For x > 5, the expression (x² − 10x + 25)/(x − 5) is equal to x − k, where k is a constant. What is the value of k? - Exponent after taking a square root
For positive x and y, √(x²y⁴) = x·yⁿ, where n is a constant. What is the value of n? - Larger of two consecutive odd integers from their product
The product of two consecutive positive odd integers is 63. What is the larger of the two integers? - Checking a radical equation for a true value
Which value of x makes the equation √(x + 4) = x − 2 a true statement? - Maximum of a quadratic from f(0)
The function f is defined by f(x) = −x² + 6x + c, where c is a constant, and f(0) = 5. What is the maximum value of f? - Percent growth over two years of an exponential model
The number of members of a club t years from now is modelled by M(t) = 500(1.2)ᵗ. By what percent does the model say the… - Product of conjugate binomials
For all values of x, (2x + 5)(2x − 5) = ax² − b, where a and b are positive constants. What is the value of a + b? - Exponent and coefficient after dividing
For x ≠ 0, the expression 12x³/(3x) is equivalent to cxⁿ, where c and n are constants. What is the value of c + n? - Sum of inputs sharing an output on a parabola
The function f is defined by f(x) = 2(x − 3)² − 5. If f(a) = f(b) = 3 and a ≠ b, what is the value of a + b? - Minimum of a factored quadratic with a leading coefficient
The function h is defined by h(x) = 3(x − 1)(x − 5). What is the minimum value of h(x)? - Value after repeated halving over a period
A quantity of 400 grams of a substance halves every 10 years. How many grams remain after 30 years? - Common factor of a difference
For all values of x, (3x)² − 9 = k(x² − 1), where k is a constant. What is the value of k? - Simplifying a quadratic over x
For x ≠ 0, the expression (x² − 5x)/x is equivalent to x − k, where k is a constant. What is the value of k? - Time to reach a maximum from vertex form
The height h, in metres, of a kite above the ground t minutes after it is launched is modelled by h(t) = −(t − 6)² + 20 … - Exponent after simplifying a product plus a constant
For all values of x, (x − 2)(x + 2) + 4 = xⁿ, where n is a constant. What is the value of n? - Rise to a maximum from a vertex-form model
The height h, in metres, of a drone t seconds after launch is modelled by h(t) = −(t − 5)² + 30 for 0 ≤ t ≤ 10. By how m… - Value after two years of 10% growth
An account holds $500 and grows by 10% each year, compounded annually. How much is in the account, in dollars, after 2 y… - Quotient of a difference of squares
For x > 0, the expression (4x² − 9)/(2x + 3) is equivalent to ax − b, where a and b are positive constants. What is the … - Exponent of a product of powers of two
For all values of x, 2ˣ · 2ˣ⁺¹ = 2ᵏ, where k is an expression in x. If x = 3, what is the value of k? - Cancelling a common binomial factor
For x > 0, the expression (x² + 2x)/(x + 2) is equivalent to which of the following? - Positive root of a quadratic with fractional solution
If 4x² + 4x − 3 = 0 and x > 0, what is the value of x? - Two numbers from a sum and a sum of squares
Two positive numbers have a sum of 7 and squares that add to 25. What is the larger of the two numbers? - Starting height from vertex form
The height h, in metres, of a thrown ball t seconds after release is modelled by h(t) = −(t − 3)² + 12. From what height… - Two years of 5% decline
A lake's fish population of 2,000 falls by 5% each year. What is the population after 2 years? - Difference between a value and the minimum
The function g is defined by g(x) = 2(x − 2)² − 5. By how much does g(5) exceed the minimum value of g? - Quotient of a difference of squares by a binomial
For x > 1, the expression (9x² − 4)/(3x − 2) is equivalent to ax + b, where a and b are constants. What is the value of … - Coefficient and exponent of a cubed monomial
The expression (3x²)³ can be written as axᵇ, where a and b are constants. What is the value of a + b? - Width of a rectangle from its diagonal
A rectangle's length is 3 units more than its width, and its diagonal is 15 units long. What is the width of the rectang… - Positive x where a line meets a parabola
In the xy-plane, the line y = 2x + 1 and the parabola y = x² − 2 intersect at two points. What is the x-coordinate of th… - Height at launch from a vertex-form model
The height h, in feet, of a firework t seconds after launch is modelled by h(t) = −(t − 4)² + 25 until it bursts at its … - Two steps of 50% growth
A colony of 1,200 ants grows by 50% each month. How many ants will there be after 2 months? - Difference of two squared monomials
For all values of x, (4x)² − (2x)² is equivalent to kx², where k is a constant. What is the value of k? - Product of two monomials
The product 2x² · 3x can be written as axⁿ. What is the value of a + n? - Positive x where two curves meet
In the xy-plane, the graphs of y = x² + 3 and y = 2x² − 1 intersect at two points. What is the x-coordinate of the inter… - Initial value from vertex form
The height h, in metres, of a jet of water t seconds after it leaves a fountain is modelled by h(t) = −(t − 2)² + 20 for… - Two years of 25% decline
A herd of 400 deer shrinks by 25% each year. How many deer will there be after 2 years? - Quotient of monomials
For x ≠ 0, the expression 8x³/(2x) is equivalent to cxⁿ. What is the value of c + n? - Sum of solutions of a quadratic set equal to a number
The equation x² + 4x − 3 = 18 has two solutions. What is their sum? - Sum of y-coordinates where a parabola meets a line
In the xy-plane, the parabola y = x² and the line x + y = 6 intersect at two points. What is the sum of the y-coordinate… - Value at zero from a known minimum
The function f is defined by f(x) = (x − 2)² + k, where k is a constant, and the minimum value of f is 5. What is the va… - Two years of 25% growth
A town's population of 1,600 grows by 25% each year. What will the population be after 2 years? - Vertex from a point on a parabola
The graph of y = x² − 6x + k in the xy-plane passes through the point (1, 2). What is the y-coordinate of the graph's ve… - Product of solutions of two separate equations
If 3x² − 12 = 0 and 2y² = 32, where x and y are both positive, what is the value of xy? - Difference of two squared radical binomials
For x > 0, which expression is equivalent to (√x + 1)² − (√x − 1)²? - Value of an expression from a ratio
If a/b = 4, where a and b are positive, what is the value of (a² − b²)/(ab)? - Base of an exponential function from two points
The graph of f(x) = a·bˣ, where a and b are positive constants, passes through the points (0, 5) and (2, 45). What is th… - Reducing a rational expression with a perfect square
For x ≠ 2 and x ≠ −2, which expression is equivalent to (x² − 4x + 4)/(x² − 4)? - Width of a rectangle from its diagonal
A rectangle has a diagonal of length 15, and its length is 3 more than its width. What is the width of the rectangle? - Reading a parabola in vertex form
The function h is defined by h(x) = −(x − 3)² + 9. Which of the following is true about the graph of h in the xy-plane? - Simplifying a difference of radicals
For x > 0, √(50x²) − √(8x²) = k·x√2, where k is a constant. What is the value of k? - Width of a uniform frame
A rectangular photograph measures 8 by 10 inches. A frame of uniform width w inches surrounds it, and the photograph and… - Property of a doubling function
The function f is defined by f(x) = 2ˣ. Which of the following is true for every value of a? - Constant completing a perfect square
If 4x² + 12x + c is equivalent to (2x + k)² for all values of x, where c and k are constants, what is the value of c? - Garden width from a path's area
A rectangular garden is twice as long as it is wide. A path 1 metre wide runs around the outside of the garden, and the … - Input of an exponential function for a given output
The function h is defined by h(x) = 3·2ˣ. For what value of x is h(x) = 96? - Simplifying a squared monomial over a monomial
For nonzero x and y, which expression is equivalent to (2x³y)² / (4x⁴y²)? - Positive solution after recognising a square
If x² + 6x + 9 = 25 and x > 0, what is the value of x? - Coefficient from the vertex's x-coordinate
The graph of y = x² + bx + 7 in the xy-plane has its vertex at x = −4. What is the value of b? - Dividing by a reciprocal
For x ≠ 0, which expression is equivalent to (3/x + 2) ÷ (1/x)? - Sum of two variables from a sum of squares equal to zero
If (x − 4)² + (y + 1)² = 0, what is the value of x + y? - Difference of squares as mental arithmetic
Using the identity a² − b² = (a − b)(a + b), what is the value of 73² − 27²? - Completing a perfect-square trinomial
The trinomial x² + bx + 49 is a perfect square for a positive constant b. What is the value of b? - Squaring a radical binomial
For x > 0, the expression (√x + 3)² can be written as x + a√x + 9. What is the value of a? - Quotient of a quadratic by its factor
The polynomial 6x² + 7x − 20 is divisible by 3x − 4. Which expression is the quotient? - Counting real linear factors
When x⁴ − 16 is factored completely over the real numbers, how many distinct linear factors appear? - x³ coefficient in a quadratic product
When (x² + 3x)(x² − x + 5) is expanded, what is the coefficient of x³? - Input at a parabola's minimum
The function g(x) = x² − 4x + 7 reaches its minimum value at x = a. What is the value of a? - Length of a vertical chord of a circle
In the xy-plane, the vertical line x = 2 crosses the circle x² + y² = 40 at two points. What is the distance between tho… - Equating exponents of a common base
If 5²ˣ = 5ˣ⁺³, what is the value of x? - Quadratic hidden in a reciprocal equation
If x − 35/x = 2 and x > 0, what is the value of x? - Square exceeding its number
A positive number's square exceeds the number itself by 56. What is the number? - Rational equation with x on both sides
If (x + 2)/(x − 1) = 3 and x ≠ 1, what is the value of x? (Enter your answer as a fraction or decimal.) - Axis of symmetry from standard form
In the xy-plane, the graph of y = 2x² − 12x + 5 is symmetric about the vertical line x = a. What is the value of a? - Shifting an exponential's input
For the function g(x) = 2ˣ, g(a) = 32 for some number a. What is the value of g(a + 2)? - Minimum from zeros and the y-intercept
A quadratic function f has zeros at x = 1 and x = 5, and f(0) = 10. What is the minimum value of f? - Four half-lives of a sample
A sample contains 80 milligrams of a substance whose amount halves every 6 hours. How many milligrams remain after 24 ho… - Coefficient from a known axis
The graph of f(x) = x² + bx + 10 is symmetric about the line x = 3. What is the value of b? - Endpoint from a given average rate
For the function f(x) = x², the average rate of change of f between x = 1 and x = c (with c > 1) is 9. What is the value…
Very hard (314)
- Converting to vertex form
The function f(x) = x² + 6x + 5 is rewritten in the form f(x) = (x + a)² + b. What is the value of b? - Solving rational equations
What is the solution to the equation 6/(x − 2) = 3? - Zeros of a polynomial
The polynomial p(x) = x³ − 4x has how many distinct real zeros? - Polynomial identities: factor theorem and remainder theorem with unknown coefficients
The polynomial p is defined by p(x) = x³ + ax² + bx − 12, where a and b are constants. If x − 2 is a factor of p(x), and… - Rewriting exponential functions to reveal a growth rate over a different time interval
The function f, defined by f(x) = 3000(1.331)^(x/12), gives the value, in dollars, of an investment x months after it wa… - Tangency condition for a line and a parabola (discriminant with a parameter)
In the xy-plane, the parabola y = x² − 4x + 7 and the line y = mx − 2, where m is a constant, intersect at exactly one p… - Determining a quadratic from its zeros and extreme value, then evaluating it
The function f is defined by f(x) = ax² + bx + c, where a, b, and c are constants. If f(2) = f(8) = 0 and the minimum va… - Completing the square
Which expression is equivalent to x² + 6x + 2? - Adding rational expressions
For x ≠ 0 and x ≠ −1, which expression is equivalent to 1/x + 1/(x + 1)? - Linear-nonlinear system
If (x, y) is a solution to the system, what is the sum of all possible values of x? - Factoring a shifted difference of squares
Which expression is equivalent to (x + 2)² − 9? - Subtracting rational expressions
For x ≠ 1 and x ≠ −1, which expression is equivalent to 2/(x − 1) − 1/(x + 1)? - Linear-quadratic system
If (x, y) is a solution to the system, what is the greatest possible value of y? - Finding the leading coefficient from the vertex and a point
The graph of a quadratic function in the xy-plane has its vertex at (2, 3) and passes through the point (4, 11). If the … - Constant that produces a repeated root
In the equation x² − 6x + c = 0, c is a constant. If the equation has exactly one real solution, what is the value of c? - Tangent line to a parabola
In the xy-plane, the line y = 2x − 1 touches the parabola y = x² at exactly one point. What is the x-coordinate of that … - Factoring completely
Which expression is equivalent to x⁴ − 16? - Rewriting a sum of squares using an identity
If (x + y)² = 25 and xy = 6, what is the value of x² + y²? - Dividing a polynomial by a monomial
For x ≠ 0, which expression is equivalent to (6x⁴ − 9x²) / (3x²)? - Coefficients from the zeros
The function f is defined by f(x) = x² + bx + c, where b and c are constants. If the zeros of f are −2 and 5, what is th… - Radical equation with an extraneous root
Which of the following gives all solutions to the equation √(3x + 1) = x − 1? - No real solutions and the discriminant
The equation x² + kx + 16 = 0, where k is a constant, has no real solutions. Which of the following must be true? - Completing the square with a leading coefficient
The expression 2x² + 12x + 5 can be written in the form 2(x + h)² + k, where h and k are constants. What is the value of… - Simplifying by factoring top and bottom
For x ≠ 3 and x ≠ −3, which expression is equivalent to (x² − x − 6)/(x² − 9)? - Range of a downward-opening parabola
The function f is defined by f(x) = 8 − (x + 1)². Which of the following describes all possible values of f(x)? - Solving an exponential model for the time
A population is modelled by P(t) = 400 · 2^(t/5), where t is the number of years from now. After how many years will the… - Sum of irrational solutions
The equation x² − 6x + 4 = 0 has two real solutions. What is the sum of the two solutions? - Product from a sum and a sum of squares
If a + b = 7 and a² + b² = 29, what is the value of ab? - Adding rational expressions with different denominators
For x ≠ −2 and x ≠ 1, which expression is equivalent to 1/(x + 2) + 3/(x − 1)? - Vertex on the x-axis
The graph of y = x² − 4x + c in the xy-plane, where c is a constant, has its vertex on the x-axis. What is the value of … - Circle and line system
If (x, y) satisfies both x² + y² = 25 and y = x + 1, what is the sum of all possible values of x? - Using a difference of squares to find a sum
If x² − y² = 40 and x − y = 4, what is the value of x + y? - Tripling period
A colony of bacteria triples in size every 2 hours. If it starts with 100 cells, how many cells will it have after 4 hou… - Two parabolas intersecting
In the xy-plane, the graphs of y = 2x² − 3 and y = x² + x + 3 intersect at two points. What is the sum of the x-coordina… - Simplifying with a difference of squares
If a ≠ b and (a² − b²)/(a − b) = 11, what is the value of a + b? - Ratio of exponential outputs
The function f is defined by f(x) = 2ˣ. For any value of x, what is the value of f(x + 3)/f(x)? - Sum of the y-coordinates of intersections
In the xy-plane, the graphs of y = x² − 4x + 3 and y = −x + 3 intersect at two points. What is the sum of the y-coordina… - Square of a sum from the sum of squares and the product
If p² + q² = 58 and pq = 21, what is the value of (p + q)²? - Range of a shifted exponential
The function f is defined by f(x) = 2ˣ + 1. Which of the following describes all possible values of f(x)? - Constant for a single intersection
In the xy-plane, the parabola y = x² + c and the line y = 2x intersect at exactly one point. What is the value of the co… - Dividing one rational expression by another
For x ≠ 0 and x ≠ −1, which expression is equivalent to (3/(x + 1)) ÷ (6/(x² + x))? - Average rate of change of a quadratic
The function f is defined by f(x) = x². What is the average rate of change of f between x = 2 and x = 5? - Sum of y-coordinates where a parabola meets a line
The parabola y = x² − 2x and the line y = x + 4 meet at points P and Q. What is the sum of the y-coordinates of P and Q? - Recognising a perfect square of a known sum
If 2x + y = 10, what is the value of 4x² + 4xy + y²? - Coefficient from the vertex's position
The function g is defined by g(x) = x² + kx + 5, where k is a constant. The vertex of the graph of g in the xy-plane has… - Coefficient for exactly one real zero
If k > 0 and the quadratic expression x² + kx + 25 has exactly one real zero, what is the value of k? - Rewriting a ratio expression
If a/b = 3, what is the value of (a + b)/b? - Other input with the same output, by symmetry
The function f is defined by f(x) = x² − 6x + 5. For what nonzero value of x is f(x) equal to f(0)? - Gap between the x-coordinates of two intersections
The parabola y = x² − 4 and the line y = 2x + 4 meet at two points. What is the distance between the x-coordinates of th… - Sum of squares from a sum of reciprocals
If x + 1/x = 5, what is the value of x² + 1/x²? - Where a quadratic is negative
The function f is defined by f(x) = x² − 2x − 8. For which values of x is f(x) < 0? - Radical equation with x on the right
If √(3x + 4) = x, what is the value of x? - Counting real fourth roots
For how many real values of x is x⁴ equal to 16? - Leading coefficient from three values
The function f is defined by f(x) = ax² + bx + c, where a, b and c are constants. If f(0) = 4, f(1) = 5 and f(−1) = 9, w… - Constant for a single real solution
For what value of the constant c does the equation 2x² + 8x + c = 0 have exactly one real solution? - Product from two squared sums
If (x + y)² = 49 and (x − y)² = 9, what is the value of xy? - Identifying exponential decay from a table
The table gives values of a function f. Which equation could define f? - Radical equation with one valid root
What is the solution to the equation √(2x + 2) = x − 3? - Factoring out a common binomial
Which expression is equivalent to 3x(x − 2) − (x − 2)²? - Zeros of a cubic
The function g is defined by g(x) = x³ − 4x. How many distinct values of x satisfy g(x) = 0? - Absolute value equation with two cases
What is the sum of all values of x that satisfy |2x − 6| = 10? - Constant from a partial-fraction style identity
For x ≠ 2, the expression (x² − 4)/(x − 2) + k is equal to x + 7 for every such x, where k is a constant. What is the va… - Reading a translated graph
The graph of y = f(x) in the xy-plane passes through the point (3, 8). The graph of y = g(x) is the graph of f shifted 2… - Discriminant and number of real solutions
For which value of c does the equation x² + 6x + c = 0 have exactly one real solution? - Constants in a combined fraction
For x > 0, the sum 1/x + 1/(x + 1) can be written as (ax + b)/(x(x + 1)), where a and b are constants. What is the value… - Interpreting an exponential model's parameters
The value V, in dollars, of a collectible t years after 2020 is modelled by V(t) = 150(1.08)ᵗ. Which statement is the be… - Number of intersection points of a line and a circle
In the xy-plane, how many points do the graphs of x² + y² = 25 and y = 7 have in common? - Counting real solutions of a radical equation
How many real solutions does the equation √(x + 7) = x − 5 have? - Simplifying a cubic over a linear factor
For x ≠ 1, which expression is equivalent to (x³ − x)/(x − 1)? - Half-life: time to reach a given amount
A sample of a substance decays so that its mass halves every 5 hours. If the sample has a mass of 400 grams now, after h… - Choosing the system that models a problem
A rectangular garden has a perimeter of 30 metres and an area of 50 square metres. Which system could be used to find it… - Which expression equals a given difference for every x
For every value of x, which expression is equal to (x + 2)² − (x − 2)²? - Finding a parameter from a graph's behaviour
The graph of f(x) = ax² + 8x + c in the xy-plane has its vertex at (2, 11). What is the value of a? - Determining k for two distinct real solutions
For which of the following values of k does the equation x² − 4x + k = 0 have two distinct real solutions? - Recognising a factored difference of cubes
Which of the following is equivalent to (x − 1)(x² + x + 1)? - Translating a graph and reading a value
The graph of y = g(x) in the xy-plane is the graph of y = x² shifted 3 units left and 2 units up. What is the value of g… - Number of solutions of a quadratic equation
How many distinct real solutions does the equation 4x² − 12x + 9 = 0 have? - Subtracting fractions with variable denominators
For x ≠ 0, which expression is equivalent to 1/x − 1/(2x)? - Solving for the input of an exponential function
The function f is defined by f(x) = 5 · 3ˣ. For what value of x does f(x) = 405? - Rational equation that becomes a quadratic
If 6/x = x − 1 and x > 0, what is the value of x? - Product of rational expressions
For x ≠ 1 and x ≠ −1, which expression is equivalent to (x² − 1)/(x + 1) · 1/(x − 1)? - Constant in an exponential model from two values
A function of the form f(x) = a · bˣ has f(0) = 6 and f(2) = 54. If b > 0, what is the value of b? - Choosing k so a line is tangent to a parabola
For which value of k does the line y = 2x + k touch the parabola y = x² at exactly one point? - Input for a given output of a quadratic
The function h is defined by h(x) = (x − 3)² + 1. For what positive value of x is h(x) = 17? - Quadratic inequality: where an expression is negative
For which values of x is x² − 6x + 8 < 0? - Coefficient of the linear term of a box's volume
A box has edge lengths x + 1, x + 2, and x + 3 centimetres, where x > 0. Its volume, in cubic centimetres, can be writte… - Solving for the base of an exponential function
A function of the form f(x) = 2 · bˣ, with b > 0, satisfies f(3) = 250. What is the value of b? - A number and its reciprocal
The sum of a positive number and its reciprocal is 5/2. What is the larger of the two possible values of the number? - Net rate as a single fraction
An inlet pipe fills a tank at a rate of 3/x tanks per hour, while a leak empties it at 1/(2x) tanks per hour, where x > … - Time for exponential growth to reach a value
A population of 250 insects doubles every 4 days. After how many days will the population first reach 4,000? - Uniform frame width from equal areas
A rectangular photograph measuring 8 cm by 12 cm is mounted in a frame of uniform width w centimetres. The area of the f… - Slope of a linear difference
The graph of y = (x + 3)(x − 3) − (x − 3)² in the xy-plane is a line. What is the slope of that line? - Input from an output of a quadratic in context
A rocket's height, in metres, t seconds after launch is h(t) = −5t² + 40t. At what time t, after reaching its peak, is t… - Choosing k so a system has no solution
For which value of k does the system y = x² + 3 and y = k have NO solution? - Time to fall below a threshold
A medicine dose of 640 milligrams in the body halves every 6 hours. After how many hours will the amount first be 40 mil… - Choosing c so a parabola and a line touch
For which value of c does the graph of y = x² + 4x + c touch the x-axis at exactly one point? - Time for repeated doubling to reach a target
A rumour is known to 3 people on day 1, and the number who know it doubles each day. On which day does the number who kn… - Choosing k for a tangent horizontal line
For which value of k does the line y = k touch the graph of y = x² − 6x + 11 at exactly one point? - Slope of a linear difference of squares
The graph of y = (x + 6)² − (x − 6)² in the xy-plane is a line. What is its slope? - Day on which growth first exceeds a threshold
A rumour is known to 5 people on day 1, and the number who know it triples each day. On which day does the number first … - Duration above a height
A ball's height, in metres, t seconds after it is thrown is h(t) = −5t² + 20t. For how many seconds is the ball higher t… - Exponential equation solved by factoring out a power
If 2^(x + 3) − 2^x = 56, what is the value of x? - Constant that makes a quadratic's minimum value zero
The function h is defined by h(x) = x² − 4x + c, where c is a constant. The minimum value of h is 0. What is the value o… - Quotient of a cubic by a known linear factor
The polynomial p(x) = x³ − 4x² + x + 6 has (x − 2) as a factor. Which expression is the quotient when p(x) is divided by… - Comparing growth over an interval for an exponential function
The function f is defined by f(x) = 4 · (1/2)^x. Which statement about f is true? - Difference of two cubes of binomials
The expression (x + 2)³ − (x − 2)³ is equivalent to ax² + b, where a and b are constants. What is the value of a + b? - Consecutive integers from the sum of their reciprocals
The sum of the reciprocals of two consecutive positive integers is 7/12. What is the smaller of the two integers? - Remainder from the remainder theorem
What is the remainder when x³ + 2x² − 5 is divided by x − 1? - Area of a rectangle from its diagonal and perimeter
A rectangle has a diagonal of length 13 and a perimeter of 34. What is the area of the rectangle? - x-intercept after a horizontal shift
The function f is defined by f(x) = x² − 2x − 8, and the function g is defined by g(x) = f(x − 3). What is the smaller o… - Quartic equation that is quadratic in x²
What is the sum of the positive solutions of x⁴ − 13x² + 36 = 0? - Rational equation with a shared denominator
For what value of x does x/(x − 3) equal 2 + 6/(x − 3)? - Difference of two compositions
The functions f and g are defined by f(x) = 2x² and g(x) = x − 1. Which expression is equal to f(g(x)) − g(f(x)) for all… - Sum of cubes from a sum and a product
If x + y = 6 and xy = 5, what is the value of x³ + y³? - Constants in the factorisation of a sum of cubes
The polynomial x³ + 8 is equal to (x + 2)(x² + bx + c) for all x, where b and c are constants. What is the value of b + … - Exponential equation that is quadratic in 2^x
The equation 2^(2x) − 5 · 2ˣ + 4 = 0 has two solutions. What is the sum of the solutions? - Area of the triangle formed by a parabola's intercepts and vertex
In the xy-plane, the graph of y = −x² + 6x − 5 has two x-intercepts and a vertex. What is the area of the triangle whose… - Coefficient in a binomial expansion
When (x + 2)⁴ is expanded, what is the coefficient of x³? - Constant that places a parabola's vertex on a given line
In the xy-plane, the vertex of the graph of y = x² − 8x + c lies on the line y = x, where c is a constant. What is the v… - Perimeter of a rectangle from its area and diagonal
A rectangle has an area of 60 square units and a diagonal of length 13 units. What is the perimeter of the rectangle? - Equation that is quadratic in a square root
The sum of a positive number and three times its square root is 10. What is the number? - Size of the corner squares cut to make a box base
Equal squares are cut from the four corners of a rectangular sheet of card measuring 20 centimetres by 30 centimetres, a… - Constant for which a line is tangent to a circle
In the xy-plane, the line y = x + k is tangent to the circle x² + y² = 8, where k is a positive constant. What is the va… - Factoring by treating a binomial as a single variable
The expression (x + 3)² − (x + 3) − 6 is equal to (x + a)(x + b) for all x, where a and b are constants with a < b. What… - Area of a right triangle from its perimeter and hypotenuse
A right triangle has a perimeter of 24 units and a hypotenuse of length 10 units. What is the area of the triangle? - Where a factored cubic is negative
The function h is defined by h(x) = (x − 1)(x − 3)(x − 5). For which values of x is h(x) less than 0? - Combining fractional powers from a known cube root
For x > 0, if x^(1/3) = 2, what is the value of x^(2/3) + x^(4/3)? - Ratio of symmetric expressions under a zero-sum condition
If a + b + c = 0 and ab + bc + ca ≠ 0, what is the value of (a² + b² + c²)/(ab + bc + ca)? - Simplifying a quotient of shifted powers
For every positive integer n, the expression (2^(n + 2) − 2ⁿ)/2^(n + 1) is equal to a constant. What is that constant? - Coefficients of a cubic from two known factors and its constant
The polynomial x³ + ax² + bx + 6 has (x − 1) and (x + 2) as factors, where a and b are constants. What is the value of a… - Squared difference of fractional powers from a reciprocal sum
For x > 0, the expression (x^(1/2) − x^(−1/2))² equals x − 2 + 1/x. If x + 1/x = 7, what is the value of (x^(1/2) − x^(−… - Difference of squares of conjugate surds
The expression (1 + √2)² − (1 − √2)² can be written as k√2, where k is an integer. What is the value of k? - Leading coefficient from three values of a quadratic
The function f is defined by f(x) = a(x − h)² + k, where a, h, and k are constants. If f(0) = f(6) = 3 and f(3) = −6, wh… - Intersection of an exponential graph and its reflection
In the xy-plane, the graphs of y = 2ˣ and y = 2⁻ˣ intersect at exactly one point. What is the y-coordinate of that point… - Counting integer exponents within a range of powers
For how many integer values of x is 3ˣ greater than or equal to 1/9 and less than or equal to 81? - Leading coefficient from zeros and a maximum value
A quadratic function f has zeros at x = 2 and x = 8, and its maximum value is 9. What is the leading coefficient of f? - Integers where a sum of absolute values attains its minimum
The function f is defined by f(x) = |x − 2| + |x + 2|. For how many integer values of x is f(x) equal to 4? - Constant of a square-root function from its domain
The function h is defined by h(x) = √(x − a), where a is a constant, and its domain is all x ≥ 3. What is the value of h… - Difference of two numbers from their sum and the sum of their squares
If x² + y² = 25 and x + y = 7, what is the value of |x − y|? - Volume of a box from its surface area and a height rule
A closed box has a square base of side s and a height of 2s. Its total surface area is 90 square units. What is the volu… - Value at zero from vertex and one point
For the quadratic function f, the graph of y = f(x) in the xy-plane is a parabola with vertex (3, −5) that passes throug… - Time for three halvings
The mass m, in grams, of a decaying sample after t days is given by m(t) = 80(1/2)^(t/5). After how many days will the m… - Yearly percentage from a doubled exponent
The function P(t) = 500(1.04)^(2t) models a population t years after 2020. By what percentage does the population increa… - Hole and asymptote of a rational function
The function r is defined by r(x) = (2x − 6)/(x² − 9). In the xy-plane, the graph of y = r(x) has a hole at x = a and a … - Quadratic from zeros and one value
The quadratic function f has zeros at x = −2 and x = 6, and f(0) = −24. What is the value of f(2)? - Where exponential overtakes linear
The functions f and g are defined by f(x) = 3x + 10 and g(x) = 2ˣ. For which of the following is g(x) greater than f(x) … - Exponential from two non-adjacent values
An exponential function h is defined by h(x) = a·bˣ, where a and b are positive constants. If h(1) = 6 and h(3) = 54, wh… - Time in the air from vertex form
The height h, in metres, of a ball t seconds after it is kicked from the ground is modelled by h(t) = −5(t − 2)² + 20. F… - Simplified rational expression constant
For x ≠ −3, the expression (x² − 9)/(x² + 6x + 9) can be written as (x − 3)/(x + k), where k is a constant. What is the … - Perfect-square trinomial constants
If 4x² − 20x + c = (2x − a)² for all values of x, where a and c are positive constants, what is the value of a + c? - Combining unlike radicals
The expression √50 + √18 can be written as k√2, where k is a constant. What is the value of k? - Exponent rules with fractional and negative powers
For x > 0, the expression x^(3/2) · x^(1/2) ÷ x^(−1) is equal to xⁿ. What is the value of n? - Counting solutions of an absolute-value equation
How many values of x satisfy |x − 4| = 2x + 1? - Tangent line meeting a parabola once
The system of equations y = x² − 4x + 3 and y = 2x − 6 has exactly one solution (x, y). What is the value of x? - Factoring a difference of cubes
The polynomial x³ − 8 can be written as (x − 2)(x² + bx + c), where b and c are constants. What is the value of b + c? - Product of x-coordinates of intersections
In the xy-plane, the circle x² + y² = 25 and the line y = x + 1 intersect at two points. What is the product of the x-co… - Simplifying a rational expression with a repeated factor
For x ≠ 2, the expression (3x² − 12)/(x² − 4x + 4) is equivalent to a(x + 2)/(x − 2), where a is a constant. What is the… - Factoring a quartic as a quadratic in x squared
For all values of x, x⁴ − 5x² + 4 = (x² − a)(x² − b), where a and b are constants with a < b. What is the value of b − a… - Simplifying a difference of powers of two
The expression (2¹⁰ − 2⁸)/2⁷ is equal to a whole number. What is that number? - Sum of coefficients from a known factor
The polynomial p(x) = x³ + ax² − 5x + b, where a and b are constants, has (x − 1) and (x + 2) as factors. What is the va… - Difference of two squared binomials
Which of the following is equivalent to (x − 2)² − (x + 2)² for all values of x? - The single root when the discriminant is zero
The equation x² − 6x + c = 0, where c is a constant, has exactly one real solution. What is that solution? - Constant for a line tangent to a parabola
The system y = x² + k and y = 2x, where k is a constant, has exactly one solution. What is the value of k? - Sum of solutions of an absolute-value equality
The equation |2x − 5| = |x + 1| has two solutions. What is their sum? - Length from a diagonal two units longer
A rectangle has a width of 4 centimetres, and its diagonal is 2 centimetres longer than its length. What is the length o… - Sum of solutions of an exponential quadratic
The equation 3^(2x) − 10·3ˣ + 9 = 0 has two solutions. What is their sum? - Leading coefficient from a maximum and a point
The quadratic function f has a maximum value of 12, which occurs at x = −1, and f(1) = 4. What is the value of the leadi… - Difference of inputs from a ratio of outputs
The function f is defined by f(x) = 3ˣ. If f(a) = 27·f(b), what is the value of a − b? - Percent decrease per period from a scaled exponent
The function P(t) = 500(0.8)^(t/3) models the amount of a substance, in grams, t hours after a measurement. By what perc… - y-coordinate of a hole in a rational graph
The function r is defined by r(x) = (x² − 1)/(x − 1) for x ≠ 1. In the xy-plane, the graph of y = r(x) is a line with a … - Product of the real solutions of a biquadratic
The equation x⁴ − 13x² + 36 = 0 has four real solutions. What is their product? - Equation with two equal radicals
What value of x satisfies √(2x − 1) = √(x + 3)? - Constant from the gap between two roots
The equation x² − 8x + c = 0, where c is a constant, has two solutions that differ by 2. What is the value of c? - Quadratic from its zeros and its peak
The quadratic function f has zeros at x = 0 and x = 6, and f(3) = 18. What is the value of f(1)? - Growth factor of an exponential from two values
The exponential function g satisfies g(0) = 4 and g(2) = 36, and g(x) is positive for every x. By what factor does g(x) … - Reducing a rational expression to a constant over x
For x ≠ 0 and x ≠ −3, the expression (2x + 6)/(x² + 3x) is equivalent to k/x, where k is a constant. What is the value o… - Constants in a factored quadratic
If (2x − 3)(ax + b) = 6x² − x − 12 for all values of x, where a and b are constants, what is the value of a + b? - Combining two rational expressions
For x ≠ 1 and x ≠ −1, the expression 1/(x − 1) − 1/(x + 1) is equivalent to k/(x² − 1), where k is a constant. What is t… - Simplifying a quotient with two variables
For nonzero a and b, which of the following is equivalent to (a²b)³/(ab²)? - Quadratic factor of a cubic
The polynomial 2x³ − 5x² − 4x + 3 has (x − 3) as a factor. Which of the following is the other factor? - Product of conjugate expressions with reciprocals
For x ≠ 0, which of the following is equivalent to (x + 1/x)(x − 1/x)? - Value of y squared at the solutions of a circle-line system
The system x² + y² = 20 and y = 2x has two solutions. What is the value of y² at either solution? - Exponential equation with two terms
If 2ˣ + 2^(x + 1) = 48, what is the value of x? - Solving a rational equation
What value of x satisfies (x − 3)/(x + 2) = 2? - Working back through a doubling rule
A function h satisfies h(x + 1) = 2·h(x) for every value of x, and h(3) = 40. What is the value of h(0)? - Vertex from intercepts
In the xy-plane, the graph of the quadratic function g crosses the x-axis at (−1, 0) and (5, 0) and the y-axis at (0, −5… - Interest earned in the second year alone
An account starts with $1,000 and earns 5% interest each year, compounded annually. How much interest, in dollars, is ea… - Transforming a function's graph
The function f is defined by f(x) = x². The graph of g in the xy-plane is the graph of f shifted 4 units to the right an… - Value after repeated percent decreases
A machine's value falls by 20% each year. It is worth $100,000 now. What will its value be, in dollars, after 3 years? - Difference of squares from a sum and a difference
If x + y = 7 and x − y = 3, what is the value of x² − y²? - Difference of squared radicals
For x > 0, the expression (√x + 1)² − (√x − 1)² is equivalent to k√x, where k is a constant. What is the value of k? - Simplifying a cubic over a quadratic
For x ≠ 1 and x ≠ −1, which of the following is equivalent to (x³ − x)/(x² − 1)? - Exponent from a product of powers of 2 and 4
If 2ᵃ · 4ᵇ = 2¹⁰ and a = 4, what is the value of b? - Constant in a sum of cubes
For all values of x, (x + 2)(x² − 2x + 4) = x³ + k, where k is a constant. What is the value of k? - Horizontal line touching a parabola once
The graphs of y = x² − 6x + 10 and y = k, where k is a constant, have exactly one point in common. What is the value of … - Constant making a line tangent to a parabola
For what value of c is the line y = x + c tangent to the parabola y = x²? - Counting solutions of a two-term absolute-value equation
How many real solutions does the equation |x − 2| + |x − 6| = 4 have? - Counting distinct real roots of a cubic
How many distinct real solutions does the equation x³ − 4x² + 4x = 0 have? - Exponential equation solved by substitution
If 9ˣ − 3ˣ − 6 = 0, what is the value of x? - Leading coefficient of a symmetric quadratic
The quadratic function q is defined by q(x) = ax² + bx + c, where a, b, and c are constants. If q(2) = q(−2), q(0) = −5,… - Zero of a reflected and shifted exponential
In the xy-plane, the graph of y = 4ˣ is reflected across the x-axis and then shifted up by 16 units. At what value of x … - Exponential growth between two readings
A bacteria population is 5 million at noon and 40 million at 3 p.m., and it grows exponentially. In millions, what was t… - Interval where a quadratic is negative
For which values of x is p(x) = x² − 8x + 7 negative? - Minimum from three symmetric points
In the xy-plane, the graph of a quadratic function passes through (0, 9), (2, 5), and (4, 9). What is the minimum value … - Y-intercept of an exponential through two points
In the xy-plane, the graph of an exponential function y = a·bˣ passes through (1, 12) and (2, 48). What is the y-interce… - Factoring a difference of shifted powers
The expression 2^(x + 2) − 2ˣ is equivalent to k·2ˣ, where k is a constant. What is the value of k? - Product of two square roots with a variable
For x > 0, the expression √x · √(4x) is equivalent to kx, where k is a constant. What is the value of k? - Dividing a polynomial by a monomial with a common factor
For x ≠ 0, which of the following is equivalent to (6x³ + 3x²)/(3x²)? - Chaining two ratios
If a/b = 4 and b/c = 3, where b and c are nonzero, what is the value of a/c? - Dividing two rational expressions
For x ≠ 2 and x ≠ −2, which of the following is equivalent to (x² − 4)/(x² + 4x + 4) ÷ (x − 2)/(x + 2)? - Sum of the squares of two roots
The equation x² − 2x − 8 = 0 has two solutions. What is the sum of their squares? - Equation with two radicals
What value of x satisfies √x + √(x + 5) = 5? - Counting solutions of an absolute-value quadratic
How many real solutions does the equation |x² − 5| = 4 have? - Positive solution of a rational equation with two fractions
What is the positive solution of 1/x + 1/(x + 1) = 3/2? - Equating powers of 4 and 2
If 4ˣ = 2^(x + 3), what is the value of x? - Maximum from two zeros and a stretch
The quadratic function f has zeros at x = 1 and x = 7, and its graph passes through (4, 9). What is the maximum value of… - Doubling model with a scaled exponent
A colony's size after t days is modelled by N(t) = 100·2^(t/3). After how many days will the colony have 800 members? - Minimum of a quadratic from its standard form
The function g is defined by g(x) = 2x² − 12x + 10. What is the minimum value of g(x)? - Amount after two years of compound interest
An investment of $2,000 grows by 10% each year, with the growth compounded annually. What is its value, in dollars, afte… - Constant in a rewritten rational expression
For all x ≠ −1, (2x − 3)/(x + 1) = 2 − k/(x + 1), where k is a constant. What is the value of k? - Square of a difference from symmetric facts
If a² + b² = 40 and ab = 12, what is the value of (a − b)²? - Number from the sum with its reciprocal
The sum of a positive number and its reciprocal is 10/3. If the number is greater than 1, what is the number? - Base of an exponential through two points
The function f is defined by f(x) = a·bˣ, where a and b are positive constants. If f(0) = 3 and f(2) = 12, what is the v… - Quadratic from two zeros and a value
A quadratic function f satisfies f(0) = 0, f(4) = 0, and f(2) = −8. What is the value of f(5)? - Sum of coefficients in a sum-of-cubes factor
For all values of x, x³ + 8 = (x + 2)(x² + bx + c), where b and c are constants. What is the value of b + c? - Rewriting a rational expression as 1 plus a fraction
For x > 2, the expression (x² − 4)/(x² − 2x) is equivalent to 1 + k/x, where k is a constant. What is the value of k? - Difference from a sum of reciprocals of powers
If 2ˣ + 2⁻ˣ = 65/8, what is the value of 2ˣ − 2⁻ˣ, given that x is positive? - Time for repeated halving
A medicine's concentration in the blood halves every 4 hours. If the concentration is 16 units at noon, at what time wil… - Sum of the zeros of a vertex-form quadratic
The function f is defined by f(x) = −(x − 4)² + 9. What is the sum of all values of x for which f(x) = 0? - Simplifying a difference over x
For x ≠ 0, the expression ((x + 1)² − 1)/x is equivalent to x + k, where k is a constant. What is the value of k? - Consecutive reciprocals summing to a fraction
The reciprocals of two consecutive positive integers have a sum of 5/6. What is the product of the two integers? - Minimum of a quadratic from its zeros
The graph of f(x) = (x − 1)(x − 5) in the xy-plane has a lowest point. What is the y-coordinate of that point? - Middle coefficients of a cubed binomial
For all values of x, (x + 2)³ = x³ + ax² + bx + 8, where a and b are constants. What is the value of a + b? - Quotient of a difference of fourth powers
For all values of x, the expression (x⁴ − 16)/(x² + 4) is equivalent to x² − k, where k is a constant. What is the value… - Perimeter of a right triangle from a relation between its sides
In a right triangle, the hypotenuse is 1 unit longer than the longer leg, and the shorter leg is 5 units. What is the pe… - Positive solution of a quadratic in a shifted variable
If (x − 1)² + (x − 1) = 6 and x > 1, what is the value of x? - Sum of a number and its reciprocal from a product
Two positive numbers have a product of 1 and a sum of 13/6. What is the larger of the two numbers? - Quotient of a trinomial by a binomial
For x > 1, the expression (2x² + 3x − 2)/(2x − 1) is equivalent to x + k, where k is a constant. What is the value of k? - Product of two numbers from their squares' sum and their ratio
Two positive numbers have squares that add to 25, and one number is 3/4 of the other. What is the product of the two num… - Side length from a quadratic relation
The area of a rectangle is 5 square units more than 4 times its width, and its length is equal to its width. What is the… - Time for a colony to reach a size
A colony of 30 insects doubles in size every 2 days. After how many days will the colony have 240 insects? - Width of a border from the enlarged area
A rectangular photograph is 10 centimetres by 15 centimetres. A border of uniform width w centimetres is added around it… - Difference between a number and its reciprocal
A number x greater than 1 satisfies x + 1/x = 10/3. What is the value of x − 1/x? - Sum of inputs with a shared output
The function g is defined by g(x) = 2(x − 4)² − 3. Two different values of x give g(x) = 15. What is the sum of those tw… - Value after two years of 10% decline
A car worth $1,000 loses 10% of its value each year. What is it worth, in dollars, after 2 years? - Coefficients of a product of three binomials
For all values of x, (x + 1)(x + 2)(x + 3) = x³ + ax² + bx + c, where a, b, and c are constants. What is the value of a … - Simplifying a cubic over a quadratic
For x > 1, the expression (x³ − x)/(x² − 1) is equivalent to xⁿ, where n is a constant. What is the value of n? - Product from the squares of a sum and a difference
If (a + b)² = 49 and (a − b)² = 9, what is the value of ab? - Product from a sum of squares and a difference
Two positive numbers have squares that add to 100, and the larger is 2 more than the smaller. What is the product of the… - Halvings needed to reach a level
A quantity starts at 64 and halves every 5 minutes. After how many minutes will it first be 8? - Minimum of a quadratic from its factored form
The function k is defined by k(x) = 3(x − 1)(x − 3). What is the minimum value of k(x)? - Total growth over four doublings
The value of a collectible is modelled by V(t) = 5 · 2ᵗ hundred dollars, where t is the number of years since purchase. … - Constant in a simplified quotient
For x > 0, the expression (3x² + 6x)/(3x) is equivalent to x + k, where k is a constant. What is the value of k? - Sum of coefficients after combining several products
For all values of x, (x + 4)(x − 4) + 16 + 2x(x − 1) = ax² + bx, where a and b are constants. What is the value of a + b… - Cube plus reciprocal cube from a sum
If x + 1/x = 5/2 and x > 1, what is the value of x³ + 1/x³? - Balance after two years of 5% interest
An account with $600 earns 5% interest each year, compounded annually. What is the balance, in dollars, after 2 years? - Rise from the minimum to a given point
The function h is defined by h(x) = 4(x − 1)² − 9. By how much does h(3) exceed the minimum value of h? - Minimum from a factored quadratic with a leading coefficient
The function p is defined by p(x) = 3(x − 2)(x − 8). What is the minimum value of p(x)? - Sum of coefficients of a quotient by x
For x ≠ 0, the expression (x³ − 4x)/x is equivalent to ax² + bx + c, where a, b, and c are constants. What is the value … - Constant left after a quotient minus x
For x ≠ 1, the expression (x² − 1)/(x − 1) − x is equal to which constant? - Legs of a right triangle from their difference and the hypotenuse
A right triangle has a hypotenuse of length 13, and one leg is 7 units longer than the other. What is the perimeter of t… - Larger solution of a squared binomial equal to a fraction
The equation (x − 1)² = 9/4 has two solutions. What is the larger one? - Gap between a value and the maximum of a parabola
The function m is defined by m(x) = −(x − 5)² + 12. By how much does the maximum value of m exceed m(2)? - Sum of two squared binomials
For all values of x, (x + 2)² + (x − 2)² = ax² + b, where a and b are constants. What is the value of a + b? - Distance between the zeros of a vertex-form quadratic
The function r is defined by r(x) = 2(x − 3)² − 8. What is the distance between the two values of x for which r(x) = 0? - Constant difference of two products
For all values of x, (x + 5)(x − 2) − (x + 1)(x + 2) is equal to which constant? - Sum of squares from a difference and a difference of squares
If a − b = 2 and a² − b² = 28, what is the value of a² + b²? - Sum of two inputs that share an output
The function m is defined by m(x) = 2(x − 4)² − 6. If m(a) = m(b) = 12 and a is not equal to b, what is the value of a +… - Difference of two numbers from their sum and sum of squares
If x + y = 8, x² + y² = 34, and x > y, what is the value of x − y? - Radical equation with an extraneous root
If √(2x + 7) = x − 4, what is the value of x? - Slope for a line tangent to a parabola
The system y = kx + 1 and y = x² + 3x + 5 has exactly one solution, and k > 0. What is the value of k? - Simplifying a rational expression to a linear one
If (x² + 5x + 6)/(x + 2) = x + c for all x ≠ −2, where c is a constant, what is the value of c? - Factoring a difference of fourth powers
Which expression is equivalent to (x⁴ − 16)/(x² + 4) for all values of x? - Adding fractional exponents
If x > 0 and x^(1/2) · x^(1/3) = xⁿ, what is the value of n? - Constant from the difference of two roots
The equation x² + 4x + c = 0 has two real solutions that differ by 6. What is the value of c? - Constant when one root is three times the other
The solutions to x² − 8x + c = 0 are p and q, where p = 3q. What is the value of c? - Time a ball spends above a height
A ball's height h, in metres, t seconds after it is launched is given by h = −5t² + 30t. For how many seconds is the bal… - Inverting a sum of reciprocals
For positive x and y, which expression is equivalent to (x⁻¹ + y⁻¹)⁻¹? - Leading coefficient from a vertex and a point
In the xy-plane, the parabola y = ax² + bx + c has its vertex at (3, −4) and passes through the point (1, 4). What is th… - Quantity halving every six hours
The amount of a substance halves every 6 hours. If there are 800 milligrams at the start, how many milligrams remain aft… - Minimum of a quadratic with a parameter
The function f is defined by f(x) = x² − 2kx + k² + 1, where k is a constant. What is the minimum value of f? - Difference of inputs from two outputs
The graph of y = 2ˣ in the xy-plane passes through the points (a, 8) and (b, 1/4). What is the value of a − b? - Sum of cubes from a sum and a product
If a + b = 7 and ab = 10, what is the value of a³ + b³? - Constant that puts a vertex on the x-axis
For what value of c does the graph of y = x² + 6x + c in the xy-plane have its vertex on the x-axis? - Value from a tripling relation
A function f satisfies f(x + 1) = 3·f(x) for all values of x, and f(2) = 18. What is the value of f(0)? - Product of a negative power and a positive power
For x ≠ 0, which of the following is equivalent to (2x)⁻² · (4x)²? - Sum of coefficients from two known solutions
The equation x² + bx + c = 0 has solutions −2 and 5, where b and c are constants. What is the value of b + c? - Leading coefficient from three points
In the xy-plane, the graph of y = ax² + bx + c passes through the points (0, 3), (1, 0), and (−1, 8). What is the value … - Product of two rational expressions
For x > 3, which expression is equivalent to (x² − 9)/(x² − 6x + 9) · (x − 3)/(x + 3)? - Counting integer solutions of a quadratic inequality
How many integer values of x satisfy x² < 10? - Minimum of a parabola from its zeros and a point
In the xy-plane, the parabola y = a(x − 2)(x − 6) passes through the point (0, 6). What is the minimum value of y on the… - Simplifying a fraction with a negative exponent
For x > 0 and x ≠ 1, which expression is equivalent to (x⁻¹ − 1)/(1 − x)? - Sum of the solutions of a second equation
The equation x² + px + q = 0 has solutions 2 and −6, where p and q are constants. If the equation x² + qx + p = 0 has so… - Height after three bounces
A ball is dropped from a height of 16 metres. After each bounce it rises to three quarters of the height from which it f… - Rewriting a power in terms of two others
If 2ᵃ = x and 2ᵇ = y, which of the following is equivalent to 2ᵃ⁺ᵇ⁺¹? - Exponential equation solved by substitution
The equation 4ˣ − 2ˣ − 2 = 0 has exactly one real solution. What is it? - Product from a sum and a difference
For numbers x and y, x + y = 9 and x − y = 5. Using the identity (x + y)² − (x − y)² = 4xy, what is the value of xy? - Subtracting rational conjugate fractions
For x ≠ 3 and x ≠ −3, the difference 1/(x − 3) − 1/(x + 3) equals a/(x² − 9) for a constant a. What is the value of a? - Collapsing a sum of equal powers
For every value of x, the sum 2ˣ + 2ˣ + 2ˣ + 2ˣ is equivalent to 2ˣ⁺ᵏ. What is the value of k? - Squared difference from sum and product
Two numbers have a sum of 9 and a product of 14. What is the square of their difference? - Fractional exponents combined
For x > 0, the expression (x^(3/2) · x^(5/2)) / x² is equivalent to xⁿ. What is the value of n? - Unknown coefficient from a known expansion
For all x, (3x − 2)(ax + 5) = 12x² + 7x − 10. What is the value of a? - Tangency condition for a parabola
In the xy-plane, the parabola y = x² + 6x + c touches the x-axis at exactly one point. What is the value of c? - Substitution for a radical quadratic
If x − √x = 6 and x > 0, what is the value of x? - Sum of squares from a nonlinear system
For numbers x and y, x + y = 10 and xy = 21. What is the value of x² + y²? - Mixed bases reduced to base two
If 2ˣ · 4ˣ⁺¹ = 8³, what is the value of x? (Enter your answer as a fraction or decimal.) - System of a product and a sum
For numbers x and y with x > y, xy = 12 and x + y = 7. What is the value of x? - Distance between a parabola's zeros
The function f(x) = (x − 2)² − 9 has two zeros. What is the distance between them on the number line? - Translating a parabola then evaluating
The graph of y = x² is shifted 3 units to the right and 2 units down. The new graph passes through the point (5, a). Wha… - Input shift as an output factor
For the functions f(x) = 2ˣ and g(x) = f(x + 3), g(x) = k · f(x) for every x. What is the value of k? - Other intercept from the vertex
A parabola has its vertex at (4, −9) and passes through (1, 0). What is the x-coordinate of its other x-intercept? - First time decay falls below half
A machine's value is 20,000 dollars and falls 20 percent each year, so its value after t years is 20,000(0.8)ᵗ. What is … - Keeping a quadratic nonnegative
What is the smallest value of c for which f(x) = x² − 10x + c is never negative? - Composition of two functions
For the functions f(x) = x² and g(x) = x + 3, what is the value of f(g(2))?
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