Bluebook SAT Test Nonlinear Equations and Systems 7 — Questions and Answers
Question 1: A quadratic equation has roots x = 2 + √3 and x = 2 − √3. What is the quadratic equation?
- x² − 4x + 1 = 0 (Correct answer)
- x² + 4x + 1 = 0
- x² − 4x − 1 = 0
- x² + 4x − 1 = 0
Correct answer: x² − 4x + 1 = 0
Sum = 4, product = (2)² − (√3)² = 4 − 3 = 1; equation: x² − 4x + 1 = 0.
Question 2: The height of a projectile is modeled by h = −5t² + 20t + 60. At what positive time t does the projectile hit the ground (h = 0)?
- t = 2 only
- t = 6 only (Correct answer)
- t = −2 and t = 6
- t = 2 and t = −6
Correct answer: t = 6 only
−5t² + 20t + 60 = 0 → t² − 4t − 12 = 0 → (t−6)(t+2) = 0 → t = 6 (positive) or t = −2 (rejected).
Question 3: If 2x² − 8 = 0, what are the solutions for x?
- x = ±2 (Correct answer)
- x = ±4
- x = ±2√2
- x = ±√2
Correct answer: x = ±2
2x² = 8 → x² = 4 → x = ±2.
Question 4: The system y = x² − x − 6 and y = 2 is solved. What is the sum of the x-coordinates of the intersection points?
- 1 (Correct answer)
- 2
- 3
- 4
Correct answer: 1
x² − x − 6 = 2 → x² − x − 8 = 0; by Vieta's formulas, sum of roots = −(−1)/1 = 1.
Question 5: What value of c makes x² + 10x + c a perfect square trinomial?
- 20
- 25 (Correct answer)
- 30
- 100
Correct answer: 25
For a perfect square: c = (b/2)² = (10/2)² = 25.
Question 6: For the system y = x² − 1 and y = 3x − 3, at which x-values do the graphs intersect?
- x = 1 and x = 2 (Correct answer)
- x = 2 and x = −1
- x = 1 only
- x = 3 and x = −1
Correct answer: x = 1 and x = 2
x² − 1 = 3x − 3 → x² − 3x + 2 = 0 → (x−1)(x−2) = 0 → x = 1 or x = 2.
Question 7: The quadratic y = −x² + 6x − 5 has a maximum value. What is that maximum value?
- 3
- 4 (Correct answer)
- 5
- 6
Correct answer: 4
Vertex x = 6/(2·1) = 3; y = −9 + 18 − 5 = 4.
Bluebook SAT Test Nonlinear Equations and Systems 7
A quadratic equation has roots x = 2 + √3 and x = 2 − √3.
What is the quadratic equation?