SAT Math Practice Test: Quadratic Equations and Functions
- Quadratic Equations and Functions
- 10 questions
- 15 minutes
These 10 questions cover quadratic equations and functions from the SAT Advanced Math domain. Decide first which form you need: factored form shows the zeros, vertex form shows the maximum or minimum.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
What is the positive solution to the equation x² − 5x − 14 = 0?
Correct answer: D
The equation factors as (x − 7)(x + 2) = 0, so x = 7 or x = −2, and the positive solution is 7. The choice 2 comes from reading the factor (x + 2) as giving x = 2.
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Question 2
f(x) = (x − 3)² + 4
What is the minimum value of the function f shown?
Correct answer: B
A square is never negative, so the smallest value of (x − 3)² is 0, reached at x = 3, and the minimum of f is 4. The choice 3 is the x-value where the minimum occurs, not the minimum itself.
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Question 3
In the equation 2x² + 4x + c = 0, c is a constant. If the equation has exactly one real solution, what is the value of c?
Correct answer: B
A quadratic has exactly one real solution when its discriminant is 0: 4² − 4(2)(c) = 16 − 8c = 0, so c = 2. The choice 4 comes from forgetting the factor 4 in the discriminant (16 − 2c · 2 = 0 is not the formula).
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Question 4
What is the product of the solutions to the equation 3x² − 12x + 5 = 0?
Correct answer: B
For ax² + bx + c = 0, the product of the solutions is c/a = 5/3. The choice 4 is the sum of the solutions, −b/a.
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Question 5
The height, in feet, of a ball t seconds after it is thrown upward is modeled by h(t) = −16t² + 64t + 5. What is the maximum height, in feet, that the ball reaches according to the model?
Correct answer: D
The vertex is at t = −64 ÷ (2 × −16) = 2, and h(2) = −64 + 128 + 5 = 69. The choice 2 is the time of the maximum, and 5 is the starting height.
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Question 6
The graph of y = x² − 2x − 8 crosses the x-axis at two points. What is the distance between these two points?
Correct answer: C
Factoring gives (x − 4)(x + 2) = 0, so the graph crosses at x = −2 and x = 4, which are 6 units apart. The choice 2 comes from subtracting 2 from 4 instead of −2 from 4.
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Question 7
Which of the following is equivalent to x² + 6x + 5 and shows the minimum value of the expression as a constant?
Correct answer: A
Completing the square: x² + 6x + 9 − 4 = (x + 3)² − 4, so the minimum value is −4. The choice (x + 3)² + 5 forgets to subtract the 9 that was added.
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Question 8
√(x + 7) = x − 5
Which of the following gives all solutions to the equation shown?
Correct answer: D
Squaring gives x + 7 = x² − 10x + 25, so x² − 11x + 18 = 0 and x = 2 or x = 9. Checking: for x = 2 the left side is 3 but the right side is −3, so 2 is extraneous and only 9 works.
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Question 9
y = x² − 4
y = 2x − 1If (a, b) is a solution to the system shown and a > 0, what is the value of b?
Correct answer: C
Setting x² − 4 = 2x − 1 gives x² − 2x − 3 = 0, so x = 3 or x = −1. With a = 3, b = 2(3) − 1 = 5; the choice −3 belongs to the solution with a = −1.
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Question 10
Which expression is equivalent to 4x² − 25?
Correct answer: A
The expression is a difference of squares, (2x)² − 5², which factors as (2x − 5)(2x + 5). The choice (2x − 5)² expands to 4x² − 20x + 25, which has a middle term.
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