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2005 AMC 12A Problem 9

Problem 9 of 25EasierAlgebra

There are two values of aa for which the equation 4x2+ax+8x+9=04x^2 + ax + 8x + 9 = 0 has only one solution for x.x. What is the sum of those values of a?a?

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Solution

The equation is 4x2+(a+8)x+9=0.4x^2 + (a+8)x + 9 = 0. It has one solution when the discriminant vanishes: (a+8)2−4⋅4⋅9=0, (a+8)^2 - 4 \cdot 4 \cdot 9 = 0, so (a+8)2=144(a+8)^2 = 144 and a+8=±12.a + 8 = \pm 12. Thus a=4a = 4 or a=−20,a = -20, and their sum is −16.-16. Thus, the correct answer is A.

More practice

Concepts: quadratic · Vieta’s Formulas

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