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 point. What is the sum of all possible values of m?

Question

Accepted Answer

B. −8. Setting the expressions equal: x² − 4x + 7 = mx − 2, so x² − (4 + m)x + 9 = 0. Exactly one intersection point means this quadratic has exactly one real solution, so its discriminant is 0: (4 + m)² − 4(1)(9) = 0, giving (4 + m)² = 36 and 4 + m = ±6. Thus m = 2 or m = −10, and the sum is −8 (choice B). (Check: m = 2 gives x² − 6x + 9 = 0, x = 3; m = −10 gives x² + 6x + 9 = 0, x = −3.) Choice C (2) keeps only the positive square root, forgetting the ± when undoing the square. Choice A (−10) keeps only the negative square root. Choice D (8) comes from expanding to m² + 8m − 20 = 0 and then reporting the sum of the roots as b/a = 8 instead of −b/a = −8.

Answer

A. −10 — You solved 4 + m = −6 correctly but stopped there — the square root gives ±6, so m = 2 is also a solution, and the question asks for the sum of all possible values.

Answer

C. 2 — You took only the positive root, 4 + m = 6, and missed 4 + m = −6, which gives m = −10; the sum of both values is −8.

Answer

D. 8 — You correctly reached m² + 8m − 20 = 0, but the sum of the roots is −b/a = −8, not b/a = 8.

Advanced Math practice question

Answer: B. −8

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