Bluebook SAT Test Nonlinear Equations and Systems 9 — Questions and Answers
Question 1: What is the sum of the roots of the equation 3x² − 12x + 7 = 0?
- 4 (Correct answer)
- 7/3
- −7/3
- −4
Correct answer: 4
By Vieta's formulas, sum of roots = −b/a = −(−12)/3 = 4.
Question 2: Which of the following equations has no real solutions?
- x² − 5x + 4 = 0
- x² + 3x + 10 = 0 (Correct answer)
- x² − 4x − 5 = 0
- x² − 6x + 9 = 0
Correct answer: x² + 3x + 10 = 0
For x² + 3x + 10 = 0, discriminant = 9 − 40 = −31 < 0, so no real solutions.
Question 3: If f(x) = x² − 6x + 8, at which x-value does the function reach its minimum?
- x = 2
- x = 3 (Correct answer)
- x = 4
- x = 6
Correct answer: x = 3
The vertex x-coordinate = −b/(2a) = 6/2 = 3.
Question 4: The parabola y = x² + 4x − 12 crosses the x-axis at two points. What is the product of those x-intercepts?
- −12 (Correct answer)
- 12
- −4
- 4
Correct answer: −12
By Vieta's formulas, product of roots = c/a = −12/1 = −12.
Question 5: Which of the following is a correct factored form of 6x² + 11x − 10?
- (2x + 5)(3x − 2) (Correct answer)
- (3x + 5)(2x − 2)
- (6x − 5)(x + 2)
- (2x − 5)(3x + 2)
Correct answer: (2x + 5)(3x − 2)
(2x + 5)(3x − 2) = 6x² − 4x + 15x − 10 = 6x² + 11x − 10. ✓
Question 6: The system y = x² − 5x + 6 and y = 0 is solved. What is the positive difference between the two solutions?
- 1 (Correct answer)
- 2
- 3
- 5
Correct answer: 1
x² − 5x + 6 = (x − 2)(x − 3) = 0; roots are x = 2 and x = 3; difference = 1.
Question 7: If (2x − 1)² = 49, what are the two values of x?
- x = 3 and x = −4
- x = 4 and x = −3 (Correct answer)
- x = 4 and x = −4
- x = 3 and x = −3
Correct answer: x = 4 and x = −3
2x − 1 = ±7; case 1: 2x = 8, x = 4; case 2: 2x = −6, x = −3.
Bluebook SAT Test Nonlinear Equations and Systems 9
What is the sum of the roots of the equation 3x² − 12x + 7 = 0?