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

The equation given is a quadratic equation of the form ax^2 + bx + c = 0. A quadratic equation has only one distinct solution when the discriminant (b^2 - 4ac) is equal to zero.

In this case, the equation is 4x^2 + ax + 8x + 9 = 0. We can rewrite this as 4x^2 + (a+8)x + 9 = 0 to match the standard form of a quadratic equation. So, a is the coefficient of x, b is (a+8), and c is 9.

Setting the discriminant equal to zero gives us:

(a+8)^2 - 449 = 0 a^2 + 16a + 64 - 144 = 0 a^2 + 16a - 80 = 0

This is a quadratic equation in a, and we can solve it using the quadratic formula a = [-b ± sqrt(b^2 - 4ac)] / (2a). In this case, a is 1, b is 16, and c is -80.

So the solutions for a are:

a = [-16 ± sqrt((16)^2 - 41(-80))] / (2*1) a = [-16 ± sqrt(256 + 320)] / 2 a = [-16 ± sqrt(576)] / 2 a = [-16 ± 24] / 2 a = -8 ± 12

So the two values of a for which the equation has only one distinct solution for x are -20 and 4. The sum of these values is -20 + 4 = -16.