Expanding and factorising quadratics | Year 11

01 / LEARN

The idea, step by step

Expand systematically

A product of two sums multiplies every term in the first bracket by every term in the second. Write the four products before collecting like terms. The middle terms are the usual source of sign errors.

The identities for a square and a difference of squares are shortcuts to the same distributive law. A squared bracket includes a middle term; it is not the sum of the two squares.

(x + a) (x + b) = x^{2} + (a + b) x + a b

Reverse a product

Factorising rewrites an expression as a product without changing its value. Remove a common factor first. For a monic quadratic, find two numbers whose sum is the coefficient of x and whose product is the constant.

For a non-monic quadratic, split the middle term into two terms whose coefficients multiply to ac. Group the resulting four terms and look for a common bracket. Expand the final product to check it.

a x^{2} + b x + c = (p x + q) (r x + s)

A common trap

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

Instead: Both brackets contain both terms: ${(x + 5)}^{2} = x^{2} + 10 x + 25$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

A negative constant

Expand and simplify $(x - 4) (x + 7)$ .

Multiply every pair: $x^{2} + 7 x - 4 x - 28$ .
Collect the linear terms: $7 x - 4 x = 3 x$ .

x^{2} + 3 x - 28

Example 2

A non-monic quadratic

Factorise $6 x^{2} + x - 2$ .

Here $a c = - 12$ . The pair 4 and -3 adds to 1.
Write $6 x^{2} + 4 x - 3 x - 2$ .
Group: $2 x (3 x + 2) - 1 (3 x + 2)$ .
Check by expansion.

(2 x - 1) (3 x + 2)

Check your understanding

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

Show answer

$(x - 4) (x + 3)$

The numbers -4 and 3 multiply to -12 and add to -1.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Expand $(x + 3) (x + 8)$ .
2 marks
Expand $(2 x - 5) (x + 4)$ .
2 marks
Factorise $x^{2} + 2 x - 35$ .
2 marks
Factorise $4 x^{2} - 25$ .
2 marks
Factorise $3 x^{2} - 12 x$ .
2 marks
Factorise $2 x^{2} - 7 x + 3$ .
3 marks

Read the worked solutions

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

Expand $(x + 3) (x + 8)$ .
2 marks
The products are $x^{2} + 8 x + 3 x + 24$ .
Collect the linear terms.
$x^{2} + 11 x + 24$
Expand $(2 x - 5) (x + 4)$ .
2 marks
Multiply: $2 x^{2} + 8 x - 5 x - 20$ .
Collect $8 x - 5 x$ .
$2 x^{2} + 3 x - 20$
Factorise $x^{2} + 2 x - 35$ .
2 marks
Find product -35 and sum 2: 7 and -5.
Write the corresponding brackets.
$(x + 7) (x - 5)$
Factorise $4 x^{2} - 25$ .
2 marks
Recognise ${(2 x)}^{2} - 5^{2}$ .
Use difference of squares.
$(2 x - 5) (2 x + 5)$
Factorise $3 x^{2} - 12 x$ .
2 marks
The common factor is 3x.
Divide both terms by it.
$3 x (x - 4)$
Factorise $2 x^{2} - 7 x + 3$ .
3 marks
Split -7x into -6x-x.
Group $2 x (x - 3) - 1 (x - 3)$ .
$(2 x - 1) (x - 3)$

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics topics 2.1–2.2: quadratic relationships, cubic and quartic polynomials. Original Ascent teaching sequence.

Official curriculum source ↗

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