Bluebook SAT Test Nonlinear Equations and Systems 1 — Questions and Answers
Question 1: What are the solutions to x² − 5x + 6 = 0?
- x = 2 and x = 3 (Correct answer)
- x = −2 and x = −3
- x = 1 and x = 6
- x = −1 and x = −6
Correct answer: x = 2 and x = 3
Factor as (x−2)(x−3) = 0, giving x = 2 and x = 3.
Question 2: The quadratic equation x² − 4x + k = 0 has exactly one real solution. What is the value of k?
- 2
- 4 (Correct answer)
- 6
- 8
Correct answer: 4
For exactly one solution the discriminant = 0: (−4)² − 4(1)(k) = 0 → 16 = 4k → k = 4.
Question 3: Which of the following are both solutions to 2x² + 3x − 2 = 0?
- x = 1/2 and x = −2 (Correct answer)
- x = 2 and x = −1/2
- x = 1 and x = −2
- x = 1/2 and x = 2
Correct answer: x = 1/2 and x = −2
Factor: (2x−1)(x+2) = 0 → x = 1/2 or x = −2.
Question 4: How many real solutions does the equation x² + 6x + 10 = 0 have?
- None (Correct answer)
- Exactly one
- Exactly two
- Infinitely many
Correct answer: None
Discriminant = 6² − 4(1)(10) = 36 − 40 = −4 < 0, so no real solutions exist.
Question 5: What is the vertex of the parabola y = x² − 8x + 15?
- (4, −1) (Correct answer)
- (4, 1)
- (−4, −1)
- (−4, 1)
Correct answer: (4, −1)
x-vertex = −(−8)/(2·1) = 4; y = 16 − 32 + 15 = −1; vertex = (4, −1).
Question 6: Solve for x: x² = 49
- x = 7 only
- x = −7 only
- x = 7 or x = −7 (Correct answer)
- No real solution
Correct answer: x = 7 or x = −7
Taking the square root of both sides gives x = ±7.
Question 7: The parabola y = x² + bx + c passes through (0, 5) and (5, 0). What is the value of b?
- −6 (Correct answer)
- −5
- −4
- −3
Correct answer: −6
From (0,5): c = 5. From (5,0): 25 + 5b + 5 = 0 → 5b = −30 → b = −6.
Bluebook SAT Test Nonlinear Equations and Systems 1
What are the solutions to x² − 5x + 6 = 0?