Bluebook SAT Test Nonlinear Equations and Systems 5 — Questions and Answers
Question 1: Using the quadratic formula, what are the solutions to 2x² − 4x − 6 = 0?
- x = 3 and x = −1 (Correct answer)
- x = 2 and x = −3
- x = 3 and x = 1
- x = −3 and x = 1
Correct answer: x = 3 and x = −1
Divide by 2: x² − 2x − 3 = 0 → (x−3)(x+1) = 0 → x = 3 or x = −1.
Question 2: The parabola y = (x − 3)² − 4 intersects the x-axis at which two values of x?
- x = 1 and x = 5 (Correct answer)
- x = 3 and x = −3
- x = 7 and x = −1
- x = 5 and x = −5
Correct answer: x = 1 and x = 5
Set y = 0: (x−3)² = 4 → x−3 = ±2 → x = 5 or x = 1.
Question 3: How many solutions does the system y = x² + 3x + 2 and y = x + 2 have?
- 0
- 1
- 2 (Correct answer)
- Infinitely many
Correct answer: 2
x² + 3x + 2 = x + 2 → x² + 2x = 0 → x(x+2) = 0 → x = 0 or x = −2; two solutions.
Question 4: The equation x² + px + 16 = 0 has two equal real roots. What is the positive value of p?
- 4
- 6
- 8 (Correct answer)
- 10
Correct answer: 8
Equal roots require discriminant = 0: p² − 64 = 0 → p = ±8; the positive value is 8.
Question 5: A parabola y = x² − 4x + c does not intersect the x-axis. Which value of c satisfies this condition?
- c = 2
- c = 4
- c = 5 (Correct answer)
- c = 1
Correct answer: c = 5
Discriminant < 0: 16 − 4c < 0 → c > 4; only c = 5 satisfies this.
Question 6: If x² − 7x + 10 = 0, what is the value of x² − 7x?
- −10 (Correct answer)
- 7
- 10
- 17
Correct answer: −10
From the equation: x² − 7x + 10 = 0 → x² − 7x = −10.
Question 7: Which of the following systems has exactly one solution?
- y = x² and y = −1
- y = x² and y = 0
- y = x² − 4 and y = −5
- y = (x−1)² and y = 0 (Correct answer)
Correct answer: y = (x−1)² and y = 0
(x−1)² = 0 → x = 1 is the only solution; all other choices yield 0 or 2 solutions.
Bluebook SAT Test Nonlinear Equations and Systems 5
Using the quadratic formula, what are the solutions to 2x² − 4x − 6 = 0?