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ACT Quadratics & factoring practice questions

Quadratics & factoring sits in the Algebra reporting category on the ACT Math section. Prep36 has 12 questions tagged to this skill, every one with a worked explanation. Three of them are below in full, with the reasoning, so you can see what the skill actually looks like before you practise it.

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What Quadratics & factoring covers on the ACT

Within this skill the Prep36 bank covers solving by factoring, vertex of a parabola, maximum value via vertex in a model, discriminant and intersections, factoring a monic quadratic and factoring with a leading coefficient.

Math is 45 questions in 50 minutes, which is about 66 seconds each, and it is one of the three sections that average into your composite. The difficulty split across the 12 questions tagged here is 4 easier, 5 medium and 3 harder.

Questions are original, written to the Enhanced ACT format. Prep36 is not affiliated with ACT, Inc.

3 quadratics & factoring questions, with the reasoning

Question 1 easy

What are the solutions to the equation x² - 5x - 14 = 0?

  1. A. x = -7 and x = 2
  2. B. x = -2 and x = 7 correct
  3. C. x = 2 and x = 7
  4. D. x = -7 and x = -2
Why: You need two numbers that multiply to -14 and add to -5, which are -7 and +2, so x² - 5x - 14 = (x - 7)(x + 2) = 0 and x = -2 and x = 7. Check x = -2: 4 + 10 - 14 = 0. The three wrong sets all flip a sign somewhere, and each fails the check: x = 2 gives 4 - 10 - 14 = -20, not 0. Tip: the roots have the opposite signs of the numbers inside the factors. If your factors are (x - 7)(x + 2), the solutions are 7 and -2, not -7 and 2.
Question 2 medium

Which expression is equivalent to (2x − 3)(x² + 4x − 1)?

  1. A. 2x³ + 5x² − 14x + 3 correct
  2. B. 2x³ + 5x² − 14x − 3
  3. C. 2x³ + 11x² − 14x + 3
  4. D. 2x³ + 5x² − 10x + 3
Why: Multiply the trinomial by each part of the binomial. The 2x gives 2x³ + 8x² − 2x, and the −3 gives −3x² − 12x + 3. Combining like terms produces 2x³ + 5x² − 14x + 3. Choice B ends in −3, but (−3)(−1) = +3, so the constant must be positive. Choice C leaves out the −3x² when combining, and choice D miscombines the x terms. Tip: check the constant term first. It is just the product of the two constants, here (−3)(−1) = 3, and that alone eliminates half the choices in seconds.
Question 3 hard

For which value(s) of k does the equation x² + kx + 9 = 0 have exactly one real solution?

  1. A. k = −3 or k = 3
  2. B. k = −6 or k = 6 correct
  3. C. k = 6 only
  4. D. k = 36 only
Why: A quadratic has exactly one real solution when the discriminant b² − 4ac equals 0. Here that means k² − 4(1)(9) = 0, so k² = 36 and k = 6 or k = −6. Choice C keeps only the positive root, but k = −6 works too, since x² − 6x + 9 = (x − 3)² has the single solution x = 3. Choice A solves k² = 9 by mistake, and choice D forgets to take the square root. Tip: k² = 36 has two answers, not one. Anytime you square-root both sides of an equation, write the plus-or-minus before you look at the choices.

The explanation is the point. Getting a question wrong tells you nothing you did not already suspect; knowing which step you missed is what changes the next one.

Practise the rest of the 12

The other 9 questions on this skill are in the free question bank, filterable by difficulty, each with the same worked explanation. No account and no card.

Open the Math bank

Frequently asked questions

How many ACT quadratics & factoring questions does Prep36 have?
12, each tagged to this skill and each with a worked explanation.
Are these real ACT questions?
No. They are original questions written to the Enhanced ACT format. Prep36 is not affiliated with ACT, Inc.
Is the Math section part of the composite?
Yes. On the Enhanced ACT the composite is the average of English, Math and Reading. Science is optional and reported separately.
Keep readingACT Math practiceEvery Math skill in one place.How many questions can you miss?The miss budget for every score.ACT Score CalculatorTurn raw scores into a composite.
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