Special binomial products | Year 9

01 / LEARN

The idea, step by step

A squared bracket is a product

${(a + b)}^{2}$ means $(a + b) (a + b)$ , not $a^{2} + b^{2}$ . Expansion gives $a^{2} + 2 a b + b^{2}$ . The middle term represents two equal cross-products. For subtraction, ${(a - b)}^{2} = a^{2} - 2 a b + b^{2}$ ; the last term is still positive.

A square of side $(x + 3)$ can be split into a square of area $x^{2}$ , two rectangles of area $3 x$ , and a square of area $9$ . This area picture explains the middle term rather than giving a rule to memorise without meaning.

Opposite signs cancel the middle

For $(a + b) (a - b)$ the cross-products are $- a b$ and $+ a b$ . They cancel, leaving $a^{2} - b^{2}$ . Use this only when the two brackets have the same first term and opposite second terms.

The identities work with numbers as well as variables. For example, $31 \times 29 = (30 + 1) (30 - 1) = 900 - 1$ . Always check that the pattern really matches before using a shortcut.

A common trap

${(x - 5)}^{2} = x^{2} - 25$ .

Instead: The squared bracket has two equal negative cross-products: $x^{2} - 10 x + 25$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example

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

Write $(x - 4) (x - 4)$ .
Expand: $x^{2} - 4 x - 4 x + 16$ .
Combine the two $- 4 x$ terms.

x^{2} - 8 x + 16

Example 2

Worked example

Calculate $52 \times 48$ without long multiplication.

$52 \times 48 = (50 + 2) (50 - 2)$ .
The difference-of-squares pattern gives $50^{2} - 2^{2}$ .
$2500 - 4 = 2496$ .

2496

Check your understanding

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

Show answer

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

The cross-products are $6 x + 6 x$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF 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 ↗

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