YEAR 9 · QUADRATICS AND PARABOLAS · 15 MIN

Factorising monic quadratics

Reverse expansion by finding two numbers with the required sum and product.

In this lesson

Factorise monic quadratic expressions
Check factorisation by expansion

Before you start

Multiplying two brackets
Signed multiplication and addition

01 / LEARN

Understand the idea.

Read both clues

A monic quadratic has coefficient $1$ on $x^{2}$ . Expanding $(x + p) (x + q)$ gives $x^{2} + (p + q) x + p q$ , so a factorisation must match two clues: the middle coefficient is the sum, and the constant is the product.

List integer factor pairs of the constant, then test their sums. For $x^{2} + 7 x + 12$ , the pair $3, 4$ fits both clues. Expanding the proposed brackets checks every coefficient rather than only the constant.

Use signs before searching

Use signs to narrow the search. A positive constant needs two numbers with matching signs, chosen to fit the middle coefficient. A negative constant needs opposite signs, with the larger magnitude controlling the sum's sign.

A factor pair that matches only the product is insufficient. Some monic quadratics have no integer factorisation, so an unsuccessful complete search does not justify inventing a pair. The examples here have integer factors.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example

Factorise $x^{2} - 9 x + 20$ .

Worked solution

1
Identify the required sum and product.
$p + q = - 9, p q = 20$
Expanding the brackets must reproduce the x-coefficient and constant.
2
Choose the signed factor pair.
$(- 4) + (- 5) = - 9, (- 4) (- 5) = 20$
A positive product and negative sum require two negative numbers.
3
Write the factors.
$x^{2} - 9 x + 20 = (x - 4) (x - 5)$
The two chosen numbers become the constants in the brackets.
4
Check by expanding.
$(x - 4) (x - 5) = x^{2} - 5 x - 4 x + 20 = x^{2} - 9 x + 20$
Expansion verifies both the middle term and constant.

Answer

$(x - 4) (x - 5)$

EXAMPLE 02

Worked example

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

Worked solution

1
Match the two coefficients.
$p + q = 1, p q = - 12$
The missing written coefficient of x is 1.
2
Find opposite-sign numbers.
$4 + (- 3) = 1, 4 (- 3) = - 12$
The negative product needs opposite signs; 4 has the larger magnitude.
3
Write and expand the factorisation.
$\begin{aligned} (x + 4) (x - 3) & = x^{2} - 3 x + 4 x - 12 \\ = x^{2} + x - 12 \end{aligned}$
The cross-products combine to the required single x-term.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

Factorise $x^{2} + 8 x + 15$ .

Show answer & reasoning

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

The pair $3, 5$ has sum $8$ and product $15$ , giving $(x + 3) (x + 5)$ . Its expansion is $x^{2} + 8 x + 15$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Factorise $x^{2} + 5 x + 6$ .
2 marks
Factorise $x^{2} - 8 x + 15$ .
2 marks
Factorise $x^{2} + 2 x - 15$ .
2 marks
Factorise $x^{2} - 4 x - 21$ .
2 marks
Factorise $x^{2} - 16$ .
2 marks
Alex writes $x^{2} + 7 x + 10 = (x + 1) (x + 10)$ . Explain the error and correct it.
3 marks

Read the worked solutions

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

Factorise $x^{2} + 5 x + 6$ .
2 marks
$2 + 3 = 5$ and $2 \times 3 = 6$ .
$(x + 2) (x + 3)$
Factorise $x^{2} - 8 x + 15$ .
2 marks
$- 3 - 5 = - 8$ and $(- 3) (- 5) = 15$ .
$(x - 3) (x - 5)$
Factorise $x^{2} + 2 x - 15$ .
2 marks
$5 - 3 = 2$ and $5 (- 3) = - 15$ .
$(x + 5) (x - 3)$
Factorise $x^{2} - 4 x - 21$ .
2 marks
$- 7 + 3 = - 4$ and $- 7 (3) = - 21$ .
$(x - 7) (x + 3)$
Factorise $x^{2} - 16$ .
2 marks
The missing $x$ coefficient is zero.
$- 4 + 4 = 0$ and $(- 4) (4) = - 16$ .
$(x - 4) (x + 4)$
Alex writes $x^{2} + 7 x + 10 = (x + 1) (x + 10)$ . Explain the error and correct it.
3 marks
$1 + 10 = 11$ , not $7$ .
$2 + 5 = 7$ and $2 \times 5 = 10$ .
$(x + 2) (x + 5)$

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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.

Read both clues

Use signs before searching

See the method in action.

Worked example

Identify the required sum and product.

Choose the signed factor pair.

Write the factors.

Check by expanding.

Worked example

Match the two coefficients.

Find opposite-sign numbers.

Write and expand the factorisation.

Try it yourself.

Check your understanding

Keep practising