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MPM2D Factoring Quadratics
Grade 10 Unit 3 — practice questions with full solutions

Common factoring, simple and complex trinomials, difference of squares and perfect squares — the skill every later quadratic question depends on.

Free MPM2D factoring quadratics practice for Grade 10 students in Ontario. These 6 questions cover the same material as a typical unit test or exam review on this topic, and every one comes with a complete worked solution — not just an answer key. Useful whether you are preparing for a MPM2D unit test, catching up on a lesson, or reviewing before the final exam.
MPM2D · Grade 10 Quadratic Relations 5 core 1 challenge
1Core
x²3x4x12x3x4The four regions together form one rectangle
The area model above represents a quadratic expression. (a) Write the total area by adding the four regions. (b) Write it as a product of two factors. (c) State what this shows.
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  1. (a) Add the regions: x² + 3x + 4x + 12 = x² + 7x + 12.
  2. (b) The whole rectangle has width (x + 3) and height (x + 4), so its area is (x + 3)(x + 4).
  3. (c) Both describe the same rectangle, so x² + 7x + 12 = (x + 3)(x + 4).
  4. This is why factoring works: you are finding the side lengths of a rectangle whose area you know.
Final answer(a) x² + 7x + 12   (b) (x + 3)(x + 4)
2Core
Factor fully:   x² + 5x − 24
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  1. Look for two numbers that multiply to −24 and add to +5.
  2. The pairs for 24 are 1&24, 2&12, 3&8, 4&6. Since the product is negative, one number is negative.
  3. +8 and −3 work: 8 × (−3) = −24 and 8 + (−3) = 5.
  4. Write the factors directly.
Final answer(x + 8)(x − 3)
3Core
Factor fully:   6x² − 5x − 6
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  1. This is a complex trinomial. Multiply a × c = 6 × (−6) = −36.
  2. Find two numbers multiplying to −36 and adding to −5: they are −9 and +4.
  3. Split the middle term: 6x² − 9x + 4x − 6.
  4. Group and factor each pair: 3x(2x − 3) + 2(2x − 3).
  5. The bracket is common, so factor it out.
Final answer(2x − 3)(3x + 2)
4Core
Factor fully:   9x² − 49
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  1. This is a difference of squares: both terms are perfect squares with a minus between them.
  2. Write each as a square: 9x² = (3x)² and 49 = 7².
  3. Apply a² − b² = (a − b)(a + b).
Final answer(3x − 7)(3x + 7)
5Core
Factor fully:   3x² − 12x − 36
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  1. Always check for a common factor first — here every term is divisible by 3.
  2. Factor it out: 3(x² − 4x − 12).
  3. Now factor the trinomial: two numbers multiplying to −12 and adding to −4 are −6 and +2.
  4. Keep the 3 in the final answer — forgetting it is the most common error here.
Final answer3(x − 6)(x + 2)
6Challenge
Factor fully:   x⁴ − 13x² + 36
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  1. Treat this as a quadratic in x². Let u = x², giving u² − 13u + 36.
  2. Two numbers multiplying to 36 and adding to −13: −4 and −9.
  3. So it factors as (u − 4)(u − 9) = (x² − 4)(x² − 9).
  4. Both brackets are differences of squares, so keep going.
  5. x² − 4 = (x−2)(x+2) and x² − 9 = (x−3)(x+3).
Final answer(x − 2)(x + 2)(x − 3)(x + 3)

MPM2D Factoring Quadratics — common questions

Short answers to the things students ask most about this unit.

What is the first step when factoring any expression?

Always look for a greatest common factor first. Pulling it out makes what remains far easier to factor, and forgetting it is the most common lost mark.

How do you factor a trinomial when the leading coefficient is not 1?

Multiply a by c, find two numbers that multiply to that product and add to b, split the middle term into those two parts, then factor by grouping.

More MPM2D practice — grade 10 academic

MPM2D Linear SystemsMPM2D Analytic GeometryMPM2D Quadratic Relations & GraphingMPM2D Solving Quadratic EquationsAll MPM2D units
← Unit 2: Analytic GeometryUnit 4: Quadratic Relations & Graphing →
Practice is step one

The classes teach the method behind every one of these.

The solutions above show the steps. The classes teach how to think about the problem in the first place — interactive, and worked through at the student’s own pace. Try 3 complete classes free — no payment required to begin.

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