Bluebook SAT Test Nonlinear Equations and Systems 4 — Questions and Answers
Question 1: The sum of a number and its square is 56. Which equation models this situation?
- x² + x − 56 = 0 (Correct answer)
- x² − x + 56 = 0
- x² + x + 56 = 0
- 2x² − 56 = 0
Correct answer: x² + x − 56 = 0
Let the number be x; x + x² = 56 rearranges to x² + x − 56 = 0.
Question 2: For the equation 4x² − 20x + 25 = 0, how many real solutions are there?
- 0
- 1 (Correct answer)
- 2
- 4
Correct answer: 1
Discriminant = (−20)² − 4(4)(25) = 400 − 400 = 0, so exactly one repeated real solution.
Question 3: What is the solution set of x² − 9 = 0?
- {3}
- {−3}
- {3, −3} (Correct answer)
- {9}
Correct answer: {3, −3}
x² = 9 → x = ±3.
Question 4: The system y = x² − 2x − 3 and y = 5 is solved. What are the x-values of the intersections?
- x = 4 and x = −2 (Correct answer)
- x = 3 and x = −2
- x = 4 and x = −1
- x = 5 and x = −1
Correct answer: x = 4 and x = −2
x² − 2x − 3 = 5 → x² − 2x − 8 = 0 → (x−4)(x+2) = 0 → x = 4 or x = −2.
Question 5: A ball is thrown upward; its height in feet at time t seconds is h(t) = −16t² + 64t. At what time does the ball return to the ground (h = 0, t > 0)?
- t = 2 s
- t = 4 s (Correct answer)
- t = 6 s
- t = 8 s
Correct answer: t = 4 s
−16t² + 64t = 0 → −16t(t−4) = 0 → t = 0 or t = 4; the ball returns at t = 4 s.
Question 6: If the graph of y = x² + bx + 4 touches the x-axis at exactly one point, which of the following could be a value of b?
- b = 2
- b = 3
- b = 4 (Correct answer)
- b = 5
Correct answer: b = 4
For one x-intercept, discriminant = 0: b² − 16 = 0 → b = ±4; so b = 4 works.
Question 7: Solve: (x + 2)(x − 5) = 0
- x = 2 and x = 5
- x = −2 and x = 5 (Correct answer)
- x = 2 and x = −5
- x = −2 and x = −5
Correct answer: x = −2 and x = 5
Set each factor to zero: x + 2 = 0 → x = −2; x − 5 = 0 → x = 5.
Bluebook SAT Test Nonlinear Equations and Systems 4
The sum of a number and its square is 56.
Which equation models this situation?