Bluebook SAT Test Nonlinear Equations and Systems 8

Question 1

Which of the following is a factor of x² + 7x + 12?

Accepted Answer

(x + 3)

Answer

x² + 7x + 12 = (x+3)(x+4), so (x+3) is a factor.

Question 2

The line y = −3x + 5 and the parabola y = x² − 4 intersect at two points. What is the product of the x-coordinates of these two points?

Accepted Answer

−9

Answer

x² − 4 = −3x + 5 → x² + 3x − 9 = 0; by Vieta's, product of roots = −9/1 = −9.

Question 3

What is the maximum value of the function f(x) = −x² + 4x + 1?

Accepted Answer

5

Answer

Vertex x = −4/(2·(−1)) = 2; f(2) = −4 + 8 + 1 = 5.

Question 4

The equation kx² − 6x + 1 = 0 has two distinct real solutions. Which of the following must be true?

Accepted Answer

k < 9

Answer

Discriminant > 0: 36 − 4k > 0 → k < 9.

Question 5

The roots of ax² + bx + c = 0 are both negative. If a > 0, which of the following must be true?

Accepted Answer

b > 0 and c > 0

Answer

Sum of roots = −b/a < 0 → b > 0; product of roots = c/a > 0 → c > 0.

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