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2005 AMC 10A Problem 10

Problem 10 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

Writing the equation as 4x2+(a+8)x+9=0,4x^2 + (a+8)x + 9 = 0, there is one solution exactly when the discriminant (a+8)2−144=0.(a+8)^2 - 144 = 0. Then a+8=±12,a + 8 = \pm 12, so a=4a = 4 or a=−20,a = -20, and their sum is −16.-16. Thus, the correct answer is A.

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Concepts: quadratic · algebraic manipulation

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.