2005 AMC 12A Problem 9

Attempt Problem 9 of the 2005 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 12A solutions, or check the answer key.

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There are two values of

a

for which the equation

4x^2 + ax + 8x + 9 = 0

has only one solution for

x.

What is the sum of those values of

a?

-16

-8

0

8

20

A quadratic has one solution exactly when its discriminant is

0

Write the linear coefficient as

a + 8

and set

(a+8)^2 - 4 \cdot 4 \cdot 9 = 0

The equation is

4x^2 + (a+8)x + 9 = 0.

It has one solution when the discriminant vanishes:

(a+8)^2 - 4 \cdot 4 \cdot 9 = 0,

(a+8)^2 = 144

and

a + 8 = \pm 12.

Thus

a = 4

a = -20,

and their sum is

-16.

Thus, the correct answer is A.

Problem 9 in other years