MCF3M Functions & Applications · Unit 2 — The Algebra of Quadratic Expressions
Review
Chapter 2 Unit Review — The Algebra of Quadratic Expressions
Find your weak areas, choose a factoring strategy, practise mixed problems and check your readiness for the Chapter 2 test.
Estimated time: 60 minutes
Learning goals
- choose a method before factoring
- factor completely and check by expansion
- connect rectangular area, dimensions and perimeter
- explain and correct errors
- identify which Chapter 2 lessons to revisit
Step 1 of 4
Warm Up
Quick Skills Check
Part 1 — Quick Skills Check
Six questions cover Lessons 2.1–2.5. Each answer explains the skill and names the lesson to revisit. Write down any skill you want to review.
Step 2 of 4
Learn
Factoring Strategy Guide
Part 2 — Factoring Strategy Guide
1. Is there a GCF? Factor it out first, then look again at what remains.
2. Two terms? Check for a difference of squares: u² − v² = (u − v)(u + v). A sum of squares is not this pattern.
3. Three terms? Check for a perfect square: u² ± 2uv + v² = (u ± v)². The middle term must fit too.
4. Leading coefficient 1? Find two integers whose sum is b and product is c.
5. Leading coefficient not 1? Use ac or guess-and-check. For ac, find a pair with product ac and sum b, split the middle and group.
6. Expand to verify all coefficients. Look for any factors you can still factor. If no integer pair works, do not force one.
Worked example
Factor completely 5x² + 25x + 30.
GCF first
Every coefficient is divisible by 5.
5(x² + 5x + 6)
Sum and product
2 and 3 have product 6 and sum 5.
5(x + 2)(x + 3)
Verify
Expand to restore every original coefficient.
5(x² + 5x + 6) = 5x² + 25x + 30
Answer: 5(x + 2)(x + 3)
Real-world idea
From area back to dimensions
A school mural is rectangular. Its area model is A = x² + 8x + 15 square metres, where x > 0 is a measurement in metres.
The factors represent length and width, not perimeter: area is their product. Changing the input x changes the two dimensions and the output area.
Worked example
Find possible dimensions and perimeter of the mural with A = x² + 8x + 15.
Factor
3 and 5 have product 15 and sum 8.
A = (x + 3)(x + 5)
Interpret
Possible dimensions are x + 3 m and x + 5 m. Both are positive for x > 0.
Perimeter
Add length and width, then double.
P = 2[(x + 3) + (x + 5)] = 4x + 16 m
Check
At x = 2, dimensions are 5 m and 7 m: area 35 m² and perimeter 24 m. The models give the same values.
Answer: Dimensions x + 3 m and x + 5 m; perimeter 4x + 16 m.
Your turn
Step 3 of 4
Practice
Mixed Practice, Applications & Find the Mistake
Part 3 — Mixed Practice
Choose your method before calculating. Work on paper, then choose the matching answer. Check every factorization by expanding.
Build it
Challenge
Part 4 — Real-World Applications
Factoring an area expression gives possible side lengths. Perimeter adds the four sides. Keep lengths positive and use linear units for lengths, square units for area.
Apply it
Application A — Garden
Apply it
Application B — Display Board
Part 5 — Find the Mistake
Read each student's work. Find the first incorrect step, choose the correction and explain why it works.
Explain it
Step 4 of 4
Mastery Check
Test Readiness Check
Part 6 — Test Readiness Check
This is practice, not the formal unit test. Answer all twelve questions before submitting. Read each explanation and use the skill report to choose your next lesson to review.
Chapter 2 Test Readiness Check
12 questions · no hints during the check. Answer them all, then check your score — you can retry as many times as you like.