SAT Math
Nonlinear Equations and Systems
Nonlinear Equations and Systems is a skill in the Advanced Math domain of the Digital SAT Math section. These are practice questions on it, and each answer choice comes with an explanation of why it is right or wrong.
13 questions. After every answer you’ll see why each choice is right or wrong, not just the one you picked.
Sample questions
Open a question to see the answer choices and the correct answer.
What is one value of x that satisfies the equation x² − 5x − 14 = 0?
- A. 5
- B. 14
- C. 2
- D. −2 ✓
Correct answer: D. The equation factors as (x − 7)(x + 2) = 0, so x = 7 or x = −2; −2 is one of the two solutions.
A ball is thrown into the air so that its height h, in feet, t seconds after it is thrown is given by h(t) = −16t² + 64t + 5. At what time t, other than t = 0, does the ball return to a height of exactly 5 feet?
- A. t = 4 ✓
- B. t = 2
- C. t = 1
- D. t = 8
Correct answer: A. Setting −16t² + 64t + 5 = 5 gives −16t² + 64t = 0, or −16t(t − 4) = 0, so t = 0 or t = 4.
Which ordered pair (x, y) is a solution to the system of equations y = x² − 6x + 8 and y = 2x − 4?
- A. (0, −4)
- B. (2, 0) ✓
- C. (4, 8)
- D. (8, 12)
Correct answer: B. Setting x² − 6x + 8 = 2x − 4 gives x² − 8x + 12 = 0, which factors as (x − 2)(x − 6) = 0; at x = 2, y = 2(2) − 4 = 0, so (2, 0) satisfies both equations.
For what positive value of k does the equation x² + kx + 25 = 0 have exactly one real solution?
- A. 10 ✓
- B. 5
- C. 25
- D. 50
Correct answer: A. Correct — a quadratic has exactly one real solution when its discriminant equals zero: k² − 4(1)(25) = 0, so k² = 100 and the positive solution is k = 10.
What is the sum of the x-coordinates of the solutions to the system y = 2x − 3 and y = x² − 4x + 5?
- A. 4
- B. 6 ✓
- C. 8
- D. 2
Correct answer: B. Correct — setting the expressions equal gives x² − 4x + 5 = 2x − 3, or x² − 6x + 8 = 0; by Vieta's formulas, the sum of the roots is −(−6)/1 = 6 (the roots are x = 2 and x = 4).