Bluebook SAT Test Nonlinear Equations and Systems 6 — Questions and Answers
Question 1: The product of two numbers is 40 and their sum is 13. Which quadratic equation has these two numbers as roots?
- x² − 13x + 40 = 0 (Correct answer)
- x² + 13x + 40 = 0
- x² − 13x − 40 = 0
- x² + 13x − 40 = 0
Correct answer: x² − 13x + 40 = 0
By Vieta's formulas: sum = 13 and product = 40 → x² − 13x + 40 = 0.
Question 2: What is the x-coordinate of the axis of symmetry of y = −2x² + 8x − 5?
- x = 2 (Correct answer)
- x = −2
- x = 4
- x = −4
Correct answer: x = 2
Axis of symmetry: x = −b/(2a) = −8 / (2·(−2)) = −8/(−4) = 2.
Question 3: The system y = 2x + k and y = x² has exactly one solution. What is k?
- −1 (Correct answer)
- −2
- 1
- 2
Correct answer: −1
x² = 2x + k → x² − 2x − k = 0; one solution when discriminant = 4 + 4k = 0 → k = −1.
Question 4: If 5x² = 45, what is the value of x²?
- 3
- 5
- 9 (Correct answer)
- 15
Correct answer: 9
x² = 45 ÷ 5 = 9.
Question 5: Which of the following best describes the solutions of x² + 2x + 5 = 0?
- Two distinct real solutions
- One repeated real solution
- No real solutions (Correct answer)
- Two equal rational solutions
Correct answer: No real solutions
Discriminant = 4 − 20 = −16 < 0, so there are no real solutions.
Question 6: The parabola y = x² − 6x + 5 is shifted 3 units to the right. What is the equation of the new parabola?
- y = x² − 12x + 32 (Correct answer)
- y = x² − 9x + 14
- y = x² − 6x + 2
- y = x² − 3x + 2
Correct answer: y = x² − 12x + 32
Replace x with (x−3): (x−3)² − 6(x−3) + 5 = x² − 6x + 9 − 6x + 18 + 5 = x² − 12x + 32.
Question 7: For the equation 2x² + 5x − 3 = 0, what is the product of the roots?
- −3/2 (Correct answer)
- 5/2
- 3/2
- −5/2
Correct answer: −3/2
By Vieta's formulas, product of roots = c/a = −3/2.
Bluebook SAT Test Nonlinear Equations and Systems 6
The product of two numbers is 40 and their sum is 13.
Which quadratic equation has these two numbers as roots?