Bluebook SAT Test Nonlinear Equations and Systems 2 — Questions and Answers
Question 1: Which values of x satisfy x² − x − 12 = 0?
- x = 4 and x = −3 (Correct answer)
- x = −4 and x = 3
- x = 6 and x = −2
- x = 3 and x = 4
Correct answer: x = 4 and x = −3
Factor: (x−4)(x+3) = 0 → x = 4 or x = −3.
Question 2: For the equation 3x² − 12x + 12 = 0, what is the value of the discriminant?
- 0 (Correct answer)
- 12
- 36
- 144
Correct answer: 0
Discriminant = (−12)² − 4(3)(12) = 144 − 144 = 0.
Question 3: Using the quadratic formula, what are the solutions of x² − 6x + 7 = 0?
- x = 3 ± √2 (Correct answer)
- x = 6 ± √2
- x = 3 ± 2√2
- x = −3 ± √2
Correct answer: x = 3 ± √2
x = [6 ± √(36−28)] / 2 = [6 ± √8] / 2 = [6 ± 2√2] / 2 = 3 ± √2.
Question 4: The system y = x² and y = x + 6 is solved by substitution. What are the x-values of the intersection points?
- x = 3 and x = −2 (Correct answer)
- x = −3 and x = 2
- x = 6 and x = −1
- x = 2 and x = −6
Correct answer: x = 3 and x = −2
Set x² = x + 6 → x² − x − 6 = 0 → (x−3)(x+2) = 0 → x = 3 or x = −2.
Question 5: How many points of intersection does the system y = x² − 3 and y = 2x have?
- 0
- 1
- 2 (Correct answer)
- 3
Correct answer: 2
x² − 3 = 2x → x² − 2x − 3 = 0; discriminant = 4 + 12 = 16 > 0, so 2 intersections.
Question 6: The function f(x) = x² − 4x + 3. At which x-value does f(x) reach its minimum?
- x = 1
- x = 2 (Correct answer)
- x = 3
- x = 4
Correct answer: x = 2
The vertex x-coordinate = 4 / (2·1) = 2; this is the minimum since a > 0.
Question 7: The system y = x + 2 and y = x² − 4 has two solutions. What is the sum of their x-coordinates?
- 1 (Correct answer)
- 2
- 3
- 5
Correct answer: 1
x² − 4 = x + 2 → x² − x − 6 = 0; by Vieta's, sum of roots = −(−1)/1 = 1.
Bluebook SAT Test Nonlinear Equations and Systems 2
Which values of x satisfy x² − x − 12 = 0?