2005 AMC 12A Problems/Problem 9

Problem

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 these values of $a$ ?

A quadratic equation always has two roots, unless it has a double root. That means we can write the quadratic as a square, and the coefficients 4 and 9 suggest this. Completing the square, $0 = (2x \pm 3)^2 = 4x^2 \pm 12x + 9$ , so $\pm 12 = a + 8 \Longrightarrow a = 4, -20$ . The sum of these is $-20 + 4 = -16 \Rightarrow \mathrm{(A)}$ .

Another method would be to use the quadratic formula, since our $x^2$ coefficient is given as 4, the $x$ coefficient is $a+8$ and the constant term is $9$ . Hence, $x = \frac{-(a+8) \pm \sqrt {(a+8)^2-4(4)(9)}}{2(4)}$ Because we want only a single solution for $x$ , the determinant must equal 0. Therefore, we can write $(a+8)^2 - 144 = 0$ which factors to $a^2 + 16a - 80 = 0$ ; using Vieta's formulas we see that the sum of the solutions for $a$ is the opposite of the coefficient of $a$ , or $-16 \Rightarrow \mathrm{ (A)}$ .