Expanding and factorising quadratics | Year 9

01 / LEARN

The idea, step by step

Multiply every term

A quadratic expression has a highest power of $2$ . In a monic quadratic such as $x^{2} + 5 x + 6$ , the coefficient of $x^{2}$ is $1$ .

To expand $(x + 2) (x + 3)$ , multiply each term in the first bracket by each term in the second. Then collect the two $x$ terms.

(x + 2) (x + 3) = x^{2} + 3 x + 2 x + 6 = x^{2} + 5 x + 6

Work backwards with a sum and a product

Factorising writes an expression as a product. To factorise $x^{2} + b x + c$ , find two integers that add to $b$ and multiply to $c$ .

If the constant is positive, the two numbers have the same sign. If it is negative, they have opposite signs. Expand your brackets to check.

x^{2} + 7 x + 12 = (x + 3) (x + 4)

Recognise useful patterns

A perfect square has a middle term: ${(x + a)}^{2} = x^{2} + 2 a x + a^{2}$ . Do not leave out $2 a x$ .

A difference of two squares has no middle term: $x^{2} - a^{2} = (x - a) (x + a)$ . These patterns are shortcuts, but you can always check by expanding.

x^{2} - 25 = (x - 5) (x + 5)

A common trap

Writing ${(x + 3)}^{2} = x^{2} + 9$ .

Instead: The two brackets give two middle terms: $(x + 3) (x + 3) = x^{2} + 3 x + 3 x + 9 = x^{2} + 6 x + 9$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Two positive brackets

Expand $(x + 4) (x + 2)$ .

Multiply every pair: $x^{2} + 2 x + 4 x + 8$ .
Collect the like terms: $2 x + 4 x = 6 x$ .

x^{2} + 6 x + 8

Example 2

Opposite signs

Factorise $x^{2} - x - 12$ .

We need a product of $- 12$ and a sum of $- 1$ . The pair is $- 4$ and $3$ .
Write $(x - 4) (x + 3)$ . Check: $x^{2} + 3 x - 4 x - 12 = x^{2} - x - 12$ .

(x - 4) (x + 3)

Example 3

A square pattern

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

The pair $- 4, - 4$ has product $16$ and sum $- 8$ .
Write $(x - 4) (x - 4)$ as ${(x - 4)}^{2}$ .

{(x - 4)}^{2}

Check your understanding

Factorise $x^{2} + 2 x - 15$ .

Show answer

$(x + 5) (x - 3)$

The numbers $5$ and $- 3$ multiply to $- 15$ and add to $2$ . Expanding the brackets gives the original expression.

03 / WORKSHEET

Your turn

6 questions · 22 marks

Worksheet PDF Worked solutions

Read the questions online

Expand and simplify.
1. (a) $(x + 3) (x + 2)$
2. (b) $(x + 4) (x - 1)$
4 marks
Factorise.
1. (a) $x^{2} + 7 x + 10$
2. (b) $x^{2} - 9 x + 20$
4 marks
Factorise.
1. (a) $x^{2} - 16$
2. (b) $x^{2} + 10 x + 25$
4 marks
Factorise $x^{2} - 3 x - 18$ . Check your answer by expanding.
3 marks
A rectangle has width $(x + 3)$ m and length $(x + 5)$ m.
1. (a)Expand an expression for its area.
2. (b)Find its area when $x = 2$ .
4 marks
Mia writes ${(x + 4)}^{2} = x^{2} + 16$ . Explain her mistake and write the correct expansion.
3 marks

Read the worked solutions

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

Expand and simplify.
1. (a) $(x + 3) (x + 2)$
2. (b) $(x + 4) (x - 1)$
4 marks
(a) Four products $x^{2} + 2 x + 3 x + 6$ [1]; collect to $x^{2} + 5 x + 6$ [1].
(b) Four products $x^{2} - x + 4 x - 4$ [1]; collect to $x^{2} + 3 x - 4$ [1].
1. (a) $x^{2} + 5 x + 6$ .
2. (b) $x^{2} + 3 x - 4$ .
Factorise.
1. (a) $x^{2} + 7 x + 10$
2. (b) $x^{2} - 9 x + 20$
4 marks
(a) Select $5, 2$ using sum and product [1]; write both brackets [1].
(b) Select $- 5, - 4$ using sum and product [1]; write both brackets [1].
1. (a) $(x + 5) (x + 2)$ .
2. (b) $(x - 5) (x - 4)$ .
Factorise.
1. (a) $x^{2} - 16$
2. (b) $x^{2} + 10 x + 25$
4 marks
(a) Recognise a difference of squares [1]; write $(x - 4) (x + 4)$ [1].
(b) Select $5, 5$ [1]; write ${(x + 5)}^{2}$ or $(x + 5) (x + 5)$ [1].
1. (a) $(x - 4) (x + 4)$ .
2. (b) ${(x + 5)}^{2}$ .
Factorise $x^{2} - 3 x - 18$ . Check your answer by expanding.
3 marks
Select $- 6, 3$ from sum $- 3$ and product $- 18$ [1].
Write $(x - 6) (x + 3)$ [1].
Check: $x^{2} + 3 x - 6 x - 18 = x^{2} - 3 x - 18$ [1].
$(x - 6) (x + 3)$
A rectangle has width $(x + 3)$ m and length $(x + 5)$ m.
1. (a)Expand an expression for its area.
2. (b)Find its area when $x = 2$ .
4 marks
(a) Use $A = (x + 3) (x + 5)$ [1]; expand to $x^{2} + 8 x + 15$ [1].
(b) Substitute $x = 2$ : $4 + 16 + 15$ or $5 \times 7$ [1]; state $35$ m $2$ [1].
1. (a) $A = x^{2} + 8 x + 15$ m $2$ .
2. (b) $35$ m $2$ .
Mia writes ${(x + 4)}^{2} = x^{2} + 16$ . Explain her mistake and write the correct expansion.
3 marks
Explain that both cross-products are needed [1].
Find $4 x + 4 x = 8 x$ [1].
Write $x^{2} + 8 x + 16$ [1].
She left out the middle term $8 x$ . The expansion is $x^{2} + 8 x + 16$ .

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Curriculum reference & lesson scope

AC9M9A02: Expand binomial products and factorise monic quadratic expressions

The quadratic part of AC9M9A02: products of two linear brackets and factorisation of monic quadratic expressions with integer coefficients. General simplification remains in Algebra. Non-monic quadratic factorisation is outside this lesson.

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