Advanced MathVery hardNo real solutions and the discriminant
Advanced Math practice question
The equation x² + kx + 16 = 0, where k is a constant, has no real solutions. Which of the following must be true?
- A. −8 < k < 8Correct
- B. k < −8 or k > 8
- C. k = 8 or k = −8
- D. k < 8
Answer: A. −8 < k < 8
No real solutions means the discriminant is negative: k² − 4(1)(16) < 0, so k² < 64, which means −8 < k < 8.
Why the other answers are wrong
- B. k < −8 or k > 8
- k² > 64 makes the discriminant positive, giving TWO real solutions.
- C. k = 8 or k = −8
- k = ±8 makes the discriminant zero, giving exactly ONE real solution.
- D. k < 8
- k < 8 includes values like k = −10, for which k² = 100 > 64 and the equation has two real solutions.
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