MCF3M Functions & Applications · Unit 2 — The Algebra of Quadratic Expressions
Lesson 2.1
Working with Quadratic Expressions
Use area models and the distributive property to expand products like (2x + 3)(x + 4) and (x − 5)² into simplified quadratic expressions.
Estimated time: 30–45 minutes
Learning goals
- multiply a monomial by a polynomial
- multiply two binomials and combine like terms
- square a binomial and explain why (x + a)² is not x² + a²
- connect polynomial multiplication to the area of a rectangle
Step 1 of 4
Warm Up
Success criteria
Real-world idea
Tiling a patio
A patio is x metres long. The owner adds 4 m to its length and 3 m to its width. How much area is there now?
You can't just multiply x by x — the extra strips add area too. Today's tool for this is the area model.
Step 2 of 4
Learn
Why the distributive property works
a(b + c) = ab + ac because a rectangle a wide and (b + c) long splits into two smaller rectangles: one ab, one ac.
Multiplying two binomials is the same idea twice: every term in the first bracket multiplies every term in the second. A 2 × 2 grid keeps track of all four products.
Key term
Expand
Rewrite a product of expressions as a sum of terms by multiplying out the brackets.
(x + 2)(x + 5) → x² + 7x + 10
Key term
Like terms
Terms with the same variable raised to the same power. Only like terms can be combined.
5x and 2x are like terms; 5x and 2x² are not.
Algebra workspace · Build the Area Model
Each row and column is a side length. Fill each region with its area, then combine like terms.
Predict first
Before you build it: how many terms will (x + 2)(x + 5) have AFTER you combine like terms?
Make a prediction to open the workspace.
About negative terms
Worked example
Expand and simplify (2x + 3)(x + 4).
First × first
2x times x gives the big region.
2x · x = 2x²
Outer
2x times 4.
2x · 4 = 8x
Inner
3 times x.
3 · x = 3x
Last × last
3 times 4.
3 · 4 = 12
Combine
8x and 3x are like terms, so add them.
2x² + 8x + 3x + 12 = 2x² + 11x + 12
Answer: 2x² + 11x + 12
Worked example
Expand (x − 5)².
Write as a product
Squaring means multiplying the binomial by itself.
(x − 5)(x − 5)
Four products
x · x, x · (−5), (−5) · x, (−5)(−5).
x² − 5x − 5x + 25
Combine
Both middle terms are −5x. They have the SAME sign, so they add instead of cancelling.
x² − 10x + 25
Answer: x² − 10x + 25. The middle term is not zero because −5x and −5x do not cancel — they double.
(x + a)² is not x² + a²
Step 3 of 4
Practice
Build it
Expand and simplify.
Apply it
Use area to model real situations.
Challenge
Think about the whole shape first.
Explain it
Explain and correct reasoning.
Step 4 of 4
Mastery Check
Lesson 2.1 mastery check
6 questions · no hints during the check. Answer them all, then check your score — you can retry as many times as you like.