Home / Assignments / MPM2D Unit 5
✎ TruMath Assignment

MPM2D Solving Quadratic Equations
Grade 10 Unit 5 — practice questions with full solutions

The quadratic formula, solving by factoring, the discriminant, and the word problems these always turn into.

Free MPM2D solving quadratic equations 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
Solve by factoring:   x² − 7x + 12 = 0
Show the full solution
  1. Factor the left side: two numbers multiplying to 12 and adding to −7 are −3 and −4.
  2. So (x − 3)(x − 4) = 0.
  3. If a product is zero, at least one factor must be zero.
  4. Set each to zero: x − 3 = 0 or x − 4 = 0.
Final answerx = 3 or x = 4
2Core
Solve using the quadratic formula, to two decimal places:   2x² + 5x − 4 = 0
Show the full solution
  1. Identify a = 2, b = 5, c = −4.
  2. The formula is x = (−b ± √(b² − 4ac)) ÷ 2a.
  3. Discriminant: 5² − 4(2)(−4) = 25 + 32 = 57.
  4. So x = (−5 ± √57) ÷ 4, and √57 ≈ 7.5498.
  5. That gives x ≈ 0.6375 and x ≈ −3.1375.
Final answerx ≈ 0.64 or x ≈ −3.14
3Core
Use the discriminant to determine how many real roots each has, without solving: (a) x² − 6x + 9 = 0   (b) 2x² + 3x + 5 = 0   (c) x² − 2x − 8 = 0
Show the full solution
  1. The discriminant is D = b² − 4ac. If D > 0 there are two real roots; if D = 0 there is one; if D < 0 there are none.
  2. (a) D = 36 − 4(1)(9) = 0, so exactly one real root (a repeated root).
  3. (b) D = 9 − 4(2)(5) = 9 − 40 = −31, which is negative, so no real roots.
  4. (c) D = 4 − 4(1)(−8) = 4 + 32 = 36, which is positive, so two real roots.
Final answer(a) one   (b) none   (c) two
4Core
The product of two consecutive positive even integers is 168. Find them.
Show the full solution
  1. Let the smaller integer be n; the next even integer is n + 2.
  2. Equation: n(n + 2) = 168, so n² + 2n − 168 = 0.
  3. Factor: two numbers multiplying to −168 and adding to 2 are 14 and −12.
  4. (n + 14)(n − 12) = 0, so n = −14 or n = 12.
  5. The question asks for positive integers, so reject −14.
Final answer12 and 14
5Core
A rectangular garden is 3 m longer than it is wide and has an area of 88 m². Find its dimensions.
Show the full solution
  1. Let the width be w; the length is w + 3.
  2. Area equation: w(w + 3) = 88, so w² + 3w − 88 = 0.
  3. Factor: numbers multiplying to −88 and adding to 3 are 11 and −8.
  4. (w + 11)(w − 8) = 0 gives w = −11 or w = 8.
  5. A width cannot be negative, so w = 8 and the length is 11.
Final answer8 m by 11 m
6Challenge
For what values of k does x² + kx + 16 = 0 have exactly one real root?
Show the full solution
  1. Exactly one real root means the discriminant equals zero.
  2. Here a = 1, b = k, c = 16, so D = k² − 4(1)(16) = k² − 64.
  3. Set D = 0: k² = 64.
  4. Taking square roots gives two answers — a very common place to lose a mark by writing only the positive one.
Final answerk = 8 or k = −8

MPM2D Solving Quadratic Equations — common questions

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

What does the discriminant tell you?

b² − 4ac shows how many real roots exist: positive gives two, zero gives one repeated root, and negative gives none.

When should I use the quadratic formula instead of factoring?

Factor first if the numbers are small and obvious. Use the formula when it does not factor neatly or when the question asks for decimal or exact radical answers.

More MPM2D practice — grade 10 academic

MPM2D Linear SystemsMPM2D Analytic GeometryMPM2D Factoring QuadraticsMPM2D Quadratic Relations & GraphingAll MPM2D units
← Unit 4: Quadratic Relations & GraphingUnit 6: Trigonometry →
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.

Try 3 Classes Free →