Bluebook SAT Test Nonlinear Equations and Systems 8 — Questions and Answers
Question 1: Which of the following is a factor of x² + 7x + 12?
- (x + 3) (Correct answer)
- (x − 3)
- (x + 6)
- (x − 4)
Correct answer: (x + 3)
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?
- −9 (Correct answer)
- −3
- 3
- 9
Correct answer: −9
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?
- 4
- 5 (Correct answer)
- 6
- 7
Correct answer: 5
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?
- k < 9 (Correct answer)
- k > 9
- k = 9
- k ≥ 9
Correct answer: k < 9
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?
- b > 0 and c > 0 (Correct answer)
- b < 0 and c > 0
- b > 0 and c < 0
- b < 0 and c < 0
Correct answer: b > 0 and c > 0
Sum of roots = −b/a < 0 → b > 0; product of roots = c/a > 0 → c > 0.
Question 6: What is the y-intercept of the parabola y = 2x² − 5x + 3?
- y = −3
- y = 0
- y = 3 (Correct answer)
- y = 5
Correct answer: y = 3
Set x = 0: y = 2(0)² − 5(0) + 3 = 3.
Question 7: Solve the system y = x + 2 and y = x² by substitution. What are the x-values of the solutions?
- x = 2 and x = −1 (Correct answer)
- x = 0 and x = 2
- x = −2 and x = 1
- x = 1 and x = −2
Correct answer: x = 2 and x = −1
x² = x + 2 → x² − x − 2 = 0 → (x−2)(x+1) = 0 → x = 2 or x = −1.
Bluebook SAT Test Nonlinear Equations and Systems 8
Which of the following is a factor of x² + 7x + 12?