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ACT · Math · Preparing for Higher Math: Algebra
Quadratic equations is part of Preparing for Higher Math: Algebra in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include solve by factoring, quadratic formula, number of real solutions, vertex, maximum and or minimum.
Common traps: factoring before setting equal to zero; reporting the factors instead of the solutions; reporting x + 3 = 0 as a root of 3 instead of -3.
Question 1
Which expression is equivalent to x² − 5x − 24?
(x − 8)(x + 3)
(x + 8)(x − 3)
(x − 6)(x + 4)
(x − 12)(x + 2)
Answer: A. (x − 8)(x + 3)
Choice A is correct. Find two numbers with product −24 and sum −5: −8 and 3. Check by FOIL: x² + 3x − 8x − 24. Choice B: has the signs swapped, giving +5x. Choice C: gives −2x. Choice D: gives −10x.
Question 2
For what value of k does the equation 2x² − 12x + k = 0 have exactly one real solution?
6
18
36
72
Answer: B. 18
A quadratic has exactly one real solution when its discriminant is zero: b² − 4ac = (−12)² − 4(2)(k) = 144 − 8k = 0, so k = 18. Choice C (36) uses 144/4, forgetting the leading coefficient 2 in 4ac; choice D (72) uses 144/2, dividing by a alone; choice A (6) is half of b, not a value of k.
Question 3
The equation kx² + 12x + 4 = 0, where k is a nonzero constant, has exactly one real solution. What is the value of k?
−9
3/4
9
36
Answer: C. 9
Exactly one real solution means b² − 4ac = 0: 144 − 4(k)(4) = 0, so 16k = 144 and k = 9. Choice A reverses the sign of 16k; B uses b instead of b² (12 − 16k = 0); D divides 144 by 4 and leaves out c = 4.
Question 4
The function f is quadratic, and its only zeros are x = 3/2 and x = −2. Which of the following could define f?
f(x) = (2x + 3)(x − 2)
f(x) = (3x − 2)(x + 2)
f(x) = (x − 3)(x + 2)
f(x) = (2x − 3)(x + 2)
Answer: D. f(x) = (2x − 3)(x + 2)
x = 3/2 makes 2x − 3 = 0 and x = −2 makes x + 2 = 0, so f(x) = (2x − 3)(x + 2) works. Choice A has zeros −3/2 and 2, both signs reversed; B has the zero 2/3, the reciprocal of 3/2; C has the zero 3, ignoring the denominator of 3/2.
Question 5
How many distinct real solutions does the equation x² - 10x + 25 = 0 have?
Exactly one
Zero
Infinitely many
Exactly two
Answer: A. Exactly one
The discriminant is b² - 4ac = (-10)² - 4(1)(25) = 0, which is zero, so there is exactly one real solution.
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