Lesson 16.2 · 25 min

Expanding Products of Brackets

Expanding $(a x + b) (c x + d)$ , products with two letters, and the perfect-square and difference-of-two-squares patterns.

In this lesson you will learn to…

Expand two brackets where $x$ has a coefficient, such as $(3 x - 4) (2 x + 6)$
Expand brackets containing different letters
Use $(a + b)^{2} = a^{2} + 2 ab + b^{2}$
Use $(a + b) (a - b) = a^{2} - b^{2}$

Why this matters

These products appear whenever you work out areas with unknown lengths and whenever you solve a quadratic. Expanding reliably — and recognising the two special patterns — makes factorising in the next lesson much easier.

Check with a number

Put $x = 2$ into the question and your answer. For $(3x - 4)(2x + 6)$ : $(2)(10) = 20$ , and $6(4) + 10(2) - 24 = 20$ ✓.

Mental maths with the patterns

$51^2 = (50 + 1)^2 = 2500 + 100 + 1 = 2601 \qquad 98 \times 102 = 100^2 - 2^2 = 9996$

Key rules and formulas

Every term by every term
$(3 x - 4) (2 x + 6) = 6 x^{2} + 18 x - 8 x - 24 = 6 x^{2} + 10 x - 24$
Four multiplications, then collect like terms. The coefficients multiply too: $3 x \times 2 x = 6 x^{2}$ .
Different letters
$(2 a + b) (c + 3 d) = 2 a c + 6 a d + b c + 3 b d$
With four different letters, there are usually no like terms to collect.
Perfect square
$(a + b)^{2} = a^{2} + 2 ab + b^{2}$
$(2 x + 3)^{2} = 4 x^{2} + 12 x + 9$ . Never just $4 x^{2} + 9$ .
Difference of two squares
$(a + b) (a - b) = a^{2} - b^{2}$
$(3 x + 5) (3 x - 5) = 9 x^{2} - 25$ . The middle terms cancel.

Try it

Area model

a (added to the width)3

b (added to the height)2

Product: (x + 3)(x + 2)
Four parts: x² + 3x + 2x + 6
Simplified: x² + 5x + 6

Each piece of the rectangle is one of the four products. The same picture works for $(a x + b) (c x + d)$ — the pieces just have different sizes.

Worked examples

Worked example 1

Coefficients of x

Expand and simplify

(3 x - 2) (4 x - 3)

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

A perfect square

Expand

(2 x - 5)^{2}

Try it yourself first, then reveal the solution one step at a time.

Practice: expanding products

8 questions

Eight questions. Type answers without brackets, e.g. 6x^2 + 10x - 24.

Question 1 of 8Algebraic answerFoundation
Expand and simplify $(2 x + 3) (x + 9)$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 2 of 8Algebraic answer
Expand and simplify $(3 x - 4) (2 x + 6)$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 3 of 8Algebraic answer
Expand and simplify $(5 x + 1) (3 x - 8)$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 4 of 8Algebraic answer
Expand $(2 p + q) (r + 3 s)$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted.
Question 5 of 8Match the pairs
Match each product to its expansion.
Pick an answer, then choose the row it belongs to. You can also drag it.
- $(2 x + 3)^{2}$
- $(2 x - 3)^{2}$
- $(2 x + 3) (2 x - 3)$
- $(3 x + 2)^{2}$
Question 6 of 8Algebraic answer
Expand $(3 x + 5) (3 x - 5)$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.
Question 7 of 8Type the answer
Use $(a + b) (a - b) = a^{2} - b^{2}$ to work out $98 \times 102$ .
Your answer
Question 8 of 8Algebraic answerChallenge
Expand and simplify $(8 x - 7) (6 x - 11)$ .
Your expression
Type powers with ^ (so n^2 is n squared). Any equivalent form is accepted, but keep it in the form the question asks for.

In this lesson you will learn to…

Why this matters

Check with a number

Mental maths with the patterns

Key rules and formulas

Every term by every term

Different letters

Perfect square

Difference of two squares

Area model

Worked examples

Practice: expanding products