ACT · Math · Preparing for Higher Math: Algebra

Quadratic equations

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.

Sample questions

Question 1
In the standard (x, y) coordinate plane, at how many points does the graph of y = 4x² − 12x + 9 meet the x-axis?
- A.
  0
- B.
  1
- C.
  2
- D.
  3
Answer: B. 1
The discriminant is b² − 4ac = (−12)² − 4(4)(9) = 144 − 144 = 0, so the parabola touches the x-axis at exactly one point, its vertex at x = 3/2. Choice A reads a zero discriminant as negative; C reads it as positive; D counts the three terms of the trinomial.
Question 2
A quadratic equation with leading coefficient 1 has solutions x = 5 and x = −3. Which of the following could be the equation?
- A.
  x² − 2x − 15 = 0
- B.
  x² + 2x − 15 = 0
- C.
  x² − 8x + 15 = 0
- D.
  x² + 8x + 15 = 0
Answer: A. x² − 2x − 15 = 0
The factors are x − 5 and x + 3, with signs opposite to the solutions, and (x − 5)(x + 3) = x² − 2x − 15. Choice B comes from (x + 5)(x − 3), using the solutions' own signs; C comes from (x − 5)(x − 3); D comes from (x + 5)(x + 3).
Question 3
For what value of k does the equation 2x² − 12x + k = 0 have exactly one real solution?
- A.
  6
- B.
  18
- C.
  36
- D.
  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 4
How many distinct real solutions does the equation x² - 10x + 25 = 0 have?
- A.
  Exactly one
- B.
  Zero
- C.
  Infinitely many
- D.
  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.
Question 5
What is the greatest real solution of the equation (x² − 3)² − 3(x² − 3) − 4 = 0?
- A.
  √2
- B.
  √7
- C.
  4
- D.
  7
Answer: B. √7
Let u = x² − 3. Then u² − 3u − 4 = 0, so (u − 4)(u + 1) = 0 and u = 4 or u = −1. That gives x² = 7 or x² = 2, so the greatest solution is √7. Choice A is the greatest solution of x² = 2; C (4) is a value of u, not x; D (7) is x², not x.

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