Lesson 2.3 · 25 min

The Product of Two Linear Expressions

Expanding $(x + a) (x + b)$ with an area model and the distributive law, including perfect squares and the difference of two squares.

In this lesson you will learn to…

Expand two brackets using an area diagram or the distributive law
Handle negative numbers inside the brackets
Recognise and expand a perfect square $(x + a)^{2}$
Recognise the difference of two squares $(x + a) (x - a)$

Why this matters

Multiplying two brackets together produces a quadratic expression — one whose highest power of $x$ is 2. Quadratics describe the path of a thrown ball, the area of a field you are fencing and the profit of a business. Being able to expand brackets reliably is the first step to working with all of them.

Seeing it as an area

A rectangle that is $(x + 2)$ wide and $(x + 3)$ tall can be split into four pieces:

	$x$	$+2$
$x$	$x^2$	$2x$
$+3$	$3x$	$6$

The total area is $x^2 + 2x + 3x + 6 = x^2 + 5x + 6$ . This grid method works just as well when the numbers are negative — you simply write $-2$ instead of $+2$ in the header.

The distributive law

Without a diagram, split the first bracket up:

$(x + 3)(x + 2) = x(x + 2) + 3(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6$

A useful memory aid is that the four products are the Firsts, Outers, Inners and Lasts.

Two special patterns

Perfect squares. $(x + 4)^2 = (x + 4)(x + 4) = x^2 + 8x + 16$ . The middle term is always double the number times $x$ :

$(x + a)^2 = x^2 + 2ax + a^2 \qquad (x - a)^2 = x^2 - 2ax + a^2$

Difference of two squares. $(x + 4)(x - 4) = x^2 - 4x + 4x - 16 = x^2 - 16$ . The middle terms cancel:

$(x + a)(x - a) = x^2 - a^2$

Mental maths trick. $49 \times 51 = (50 - 1)(50 + 1) = 50^2 - 1^2 = 2499$ . The difference of two squares turns a tricky product into an easy one.

Always check with a number

After expanding, substitute a simple value such as $x = 2$ into both the brackets and your answer. If they do not give the same number, something has gone wrong.

Key rules and formulas

Every term times every term
$(x + 3) (x + 2) = x^{2} + 2 x + 3 x + 6 = x^{2} + 5 x + 6$
Each term in the first bracket multiplies each term in the second — four multiplications in total. Then collect the two $x$ terms.
The general pattern
$(x + a) (x + b) = x^{2} + (a + b) x + ab$
The $x$ coefficient is the sum of the two numbers; the constant is their product. This is the key to factorising quadratics later.
Perfect squares
$(x + a)^{2} = x^{2} + 2 a x + a^{2}$
Squaring a bracket means multiplying it by itself. $(x + 3)^{2}$ is not $x^{2} + 9$ — you lose the middle term.
Difference of two squares
$(x + a) (x - a) = x^{2} - a^{2}$
The two middle terms, $+ a x$ and $- a x$ , cancel out, leaving no $x$ term at all.

Try it

Area model for two brackets

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

The big rectangle is $(x + a)$ wide and $(x + b)$ tall. Its area is the four smaller pieces added together. Move the sliders and watch the middle term always come out as $(a + b) x$ .

Worked examples

Worked example 1

Using the distributive law

Expand and simplify

(x + 7) (x - 2)

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

Worked example 2

Two negatives

Expand and simplify

(x - 7) (x - 3)

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

Worked example 3

A perfect square

Fatima says

(x + 3)^{2} = x^{2} + 9

. Aisha says

(x + 3)^{2} = x^{2} + 6 x + 9

. Who is right?

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

Practice: expanding two brackets

11 questions

Ten questions. When you are asked to expand, type your answer without brackets, e.g. x^2 + 5x + 6.

Question 1 of 11Complete the tableFoundation
Complete the grid to expand $(x + 5) (x + 1)$ .
× x +5
x Row 1, x Row 1, +5
+1 Row 2, x Row 2, +5
Question 2 of 11Algebraic answer
Expand and simplify $(x + 3) (x + 8)$ .
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 3 of 11Algebraic answer
Expand and simplify $(x + 8) (x - 7)$ .
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 11Algebraic answer
Expand and simplify $(x - 9) (x - 7)$ .
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 5 of 11Explore and match
Set the sliders so the rectangle has area $x^{2} + 10 x + 25$ .
Move the controls until you match the question, then press Check.
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
Question 6 of 11Multiple choice
Fatima wrote $(x + 3)^{2} = x^{2} + 9$ . What has she done wrong?
Question 7 of 11Match the pairs
Match each product to its expansion.
Pick an answer, then choose the row it belongs to. You can also drag it.
- $(x + 5)^{2}$
- $(x - 5)^{2}$
- $(x + 5) (x - 5)$
- $(x + 1) (x + 25)$
Question 8 of 11Fill the gaps
Complete these using the patterns for perfect squares and the difference of two squares.
$(x + 9) (x - 9) = x^{2} - □$ : Gap a $(x + 12)^{2} = x^{2} + □ x + 144$ : Gap b $x^{2} - 36 = (x + □) (x - 6)$ : Gap c $x^{2} + 20 x + 100 = (x + □)^{2}$ : Gap d
Question 9 of 11Type the answer
Use $(50 - 1) (50 + 1) = 5 0^{2} - 1^{2}$ to work out $49 \times 51$ without a calculator.
Your answer
Question 10 of 11Algebraic answer
A rectangle is $(x + 7)$ long and $(x - 4)$ wide. Write its area in the form $x^{2} + a x + b$ .
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 11 of 11Quick-fire drill
Quick-fire: the middle term. Type just the coefficient of $x$ (e.g. -3) when each product is expanded.
8 questions · 75 seconds · answer as many as you can.

×	x	+5
x	Row 1, x	Row 1, +5
+1	Row 2, x	Row 2, +5

In this lesson you will learn to…

Why this matters

Seeing it as an area

The distributive law

Two special patterns

Always check with a number

Key rules and formulas

Every term times every term

The general pattern

Perfect squares

Difference of two squares

Area model for two brackets

Worked examples

Practice: expanding two brackets