Advanced MathHardNumber of solutions of a tangent system
Advanced Math practice question
How many solutions does the system y = x² and y = 6x − 9 have?
- A. Exactly oneCorrect
- B. Exactly two
- C. None
- D. Infinitely many
Answer: A. Exactly one
Set equal: x² − 6x + 9 = (x − 3)² = 0, one repeated root x = 3. The line is tangent to the parabola at (3, 9).
Why the other answers are wrong
- B. Exactly two
- Two solutions would need two distinct roots; the discriminant 36 − 36 = 0 gives one.
- C. None
- x = 3 does satisfy both equations (9 = 9).
- D. Infinitely many
- A line and a parabola can't coincide.
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