Math: 3d12b1e0

ID: 3d12b1e0

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Advanced Math)

Comment: I cannot say it enough: know how to work with the discriminant—the part under the root within the quadratic formula, b^2 - 4ac—to derive no solutions, one solution, or two solutions.

Method 1: If there is one solution to the quadratic presented, then we can set the discriminant equal to 0 and solve.

b^2 - 4ac = 0

(-8)^2 - 4(-16)c = 0

64 + 64c = 0

c = -1

The correct answer is -1.

Method 2: Alter the equation slightly for Desmos, add a slider for c, and look for where the function equals 0 (or touches the x-axis).

When you slide to the right, the graph shifts upward; when you slide to the left, you land upon the correct answer pretty quickly. When c = -1, the parabola touches the x-axis, indicating a single solution. Thus, the correct answer is -1. (Remember, the question is not asking for the value of x, but of c.)

Method 3: If you are savvy with quadratics, then you can quickly determine that this one must be a perfect square (following the form y = (ax - b)^2 or y = (ax + b)^2) to have one solution. Divide out the coefficient of the x^2 term and work with such an understanding.

-16x^2 - 8x + c = 0

x^2 + (1/2)x - (c/16) = 0

Now, ignore (c/16) for a moment. For x^2 + (1/2)x to be a complete square, it must be true that the factored form is 1/4. Why? You could complete the square, but you can also see that 

(x + (1/4))^2 = x^2 + (1/4)x + (1/4)x + 1/16

x^2 + (1/2)x + 1/16

Notice that x^2 and (1/2)x match the first two terms of the altered quadratic from earlier, after dividing out the -16. Thus, the following must be true:

1/16 = -c/16

1 = -c

-1 = c

The correct answer is -1.