YEAR 9 · QUADRATICS AND PARABOLAS · 15 MIN

Special binomial products

Recognise perfect squares and differences of squares without losing the cross-products.

In this lesson

Expand squared binomials
Recognise a difference of squares

Before you start

Multiplying two brackets

01 / LEARN

Understand the idea.

A squared bracket is a product

A squared bracket means multiplying the bracket by itself. Each term in the first bracket multiplies each term in the second. The two cross-products combine to give the middle term $2 a b$ in ${(a + b)}^{2}$ .

For ${(a - b)}^{2}$ , both cross-products are negative, while $(- b) (- b)$ is positive. The area split for ${(x + 3)}^{2}$ shows why the middle term appears: two rectangles contribute $3 x$ each, alongside the two squares.

Opposite signs cancel the middle

In $(a + b) (a - b)$ , the cross-products are $- a b$ and $a b$ , so they cancel. This difference of squares pattern requires matching first terms and opposite second terms; it does not describe a squared sum or difference.

Use the identity to simplify numerical products around a convenient middle number. Check that the numbers are equally far above and below that middle before replacing the product with the difference of two squares.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

Expand ${(x - 4)}^{2}$ .

Worked solution

1
Write the square as a product.
${(x - 4)}^{2} = (x - 4) (x - 4)$
The exponent applies to the whole bracket.
2
Multiply all four term pairs.
$(x - 4) (x - 4) = x^{2} - 4 x - 4 x + 16$
Both cross-products are negative; the product of the two negative constants is positive.
3
Collect like terms.
$x^{2} - 4 x - 4 x + 16 = x^{2} - 8 x + 16$
The two x-terms have the same power and combine.

Answer

$x^{2} - 8 x + 16$

EXAMPLE 02

Worked example

Calculate $52 \times 48$ without long multiplication.

Worked solution

1
Choose the middle number.
$52 \times 48 = (50 + 2) (50 - 2)$
Both factors are 2 away from 50 with opposite signs.
2
Apply the difference of squares.
$(50 + 2) (50 - 2) = 50^{2} - 2^{2}$
The positive and negative cross-products cancel.
3
Evaluate the two squares.
$2500 - 4 = 2496$
Subtract the small square from the convenient square of 50.

Answer

$2496$

03 / YOUR TURN

Try it yourself.

Check your understanding

Expand ${(x + 6)}^{2}$ .

Show answer & reasoning

$x^{2} + 12 x + 36$

Expand $(x + 6) (x + 6)$ : the cross-products are $6 x$ and $6 x$ , while the constant product is $36$ . Thus ${(x + 6)}^{2} = x^{2} + 12 x + 36$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Expand ${(x + 3)}^{2}$ .
2 marks
Expand ${(x - 7)}^{2}$ .
2 marks
Expand ${(2 x + 1)}^{2}$ .
2 marks
Expand $(x + 8) (x - 8)$ .
2 marks
Use a difference of squares to calculate $41 \times 39$ .
3 marks
Explain why ${(x + 2)}^{2}$ and $(x + 2) (x - 2)$ are different.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Expand ${(x + 3)}^{2}$ .
2 marks
$x^{2} + 3 x + 3 x + 9 = x^{2} + 6 x + 9$ .
$x^{2} + 6 x + 9$
Expand ${(x - 7)}^{2}$ .
2 marks
$x^{2} - 7 x - 7 x + 49 = x^{2} - 14 x + 49$ .
$x^{2} - 14 x + 49$
Expand ${(2 x + 1)}^{2}$ .
2 marks
${(2 x)}^{2} + 2 (2 x) (1) + 1^{2} = 4 x^{2} + 4 x + 1$ .
$4 x^{2} + 4 x + 1$
Expand $(x + 8) (x - 8)$ .
2 marks
The $8 x$ and $- 8 x$ terms cancel.
$x^{2} - 64$
Use a difference of squares to calculate $41 \times 39$ .
3 marks
$(40 + 1) (40 - 1) = 40^{2} - 1^{2} = 1600 - 1$ .
$1599$
Explain why ${(x + 2)}^{2}$ and $(x + 2) (x - 2)$ are different.
3 marks
The first has equal signs, so the middle terms add.
The second has opposite signs, so the middle terms cancel.
$x^{2} + 4 x + 4$ and $x^{2} - 4$

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

AC9M9A02, AC9M9A04–A06: monic quadratic expressions, integer-root equations, graphs and models.

Official curriculum source ↗

In this lesson

Understand the idea.

A squared bracket is a product

Opposite signs cancel the middle

See the method in action.

Worked example

Write the square as a product.

Multiply all four term pairs.

Collect like terms.

Worked example

Choose the middle number.

Apply the difference of squares.

Evaluate the two squares.

Try it yourself.

Check your understanding

Keep practising