Solving by factorisation | Year 11

01 / LEARN

The idea, step by step

Zero is essential

If a product is zero, at least one factor is zero. This applies to $(x - 2) (x + 5) = 0$ , but not to a product equal to 6. Rearrange the equation before factorising.

Each linear factor gives a possible solution. Check in the original equation, especially when a context only permits positive lengths or times.

A B = 0 \Rightarrow A = 0 or B = 0

Do not lose a factor

For $x^{2} = 4 x$ , dividing immediately by x loses the valid solution zero. Instead write $x (x - 4) = 0$ .

A repeated factor gives one distinct root. For ${(x - 3)}^{2} = 0$ , the only solution is x=3, with multiplicity two.

A common trap

Cancelling x in $x (x - 4) = 0$ and keeping only x=4.

Instead: Cancellation assumes x is nonzero. The factor x also gives x=0.

02 / WORKED EXAMPLES

See the thinking

Example 1

Rearrange first

Solve $x^{2} + 2 x = 15$ .

Write $x^{2} + 2 x - 15 = 0$ .
Factor: $(x + 5) (x - 3) = 0$ .
Set each factor equal to zero.

x = - 5

x = 3

Example 2

Keep zero

Solve $3 x^{2} = 6 x$ .

Subtract 6x: $3 x^{2} - 6 x = 0$ .
Factor $3 x (x - 2) = 0$ .
Use each factor.

x = 0

x = 2

Check your understanding

Solve $(2 x + 1) (x - 6) = 0$ .

Show answer

$x = - \frac{1}{2}$ or $x = 6$

Solve each linear factor separately.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Solve $x^{2} - 9 x + 20 = 0$ .
2 marks
Solve $2 x^{2} + 7 x + 3 = 0$ .
2 marks
Solve $x^{2} = 7 x$ .
2 marks
Solve $4 x^{2} - 25 = 0$ .
2 marks
Solve $x^{2} + 6 x + 9 = 0$ .
2 marks
A rectangle has sides x cm and (x+2) cm and area 48 square centimetres. Find x.
3 marks

Read the worked solutions

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

Solve $x^{2} - 9 x + 20 = 0$ .
2 marks
Factor $(x - 4) (x - 5) = 0$ .
Use each factor.
$x = 4, 5$
Solve $2 x^{2} + 7 x + 3 = 0$ .
2 marks
Factor $(2 x + 1) (x + 3) = 0$ .
Solve both equations.
$x = - 3, - \frac{1}{2}$
Solve $x^{2} = 7 x$ .
2 marks
Rearrange and factor $x (x - 7) = 0$ .
Keep both factors.
$x = 0, 7$
Solve $4 x^{2} - 25 = 0$ .
2 marks
Factor $(2 x - 5) (2 x + 5) = 0$ .
Solve both factors.
$x = \pm \frac{5}{2}$
Solve $x^{2} + 6 x + 9 = 0$ .
2 marks
Factor ${(x + 3)}^{2} = 0$ .
There is one distinct root.
$x = - 3$ , repeated.
A rectangle has sides x cm and (x+2) cm and area 48 square centimetres. Find x.
3 marks
Write $x (x + 2) = 48$ , so $(x + 8) (x - 6) = 0$ .
Reject the negative length -8.
$x = 6$ cm

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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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