YEAR 9 · QUADRATICS AND PARABOLAS · 15 MIN

Expanding two brackets

Multiply every term in one bracket by every term in the other, then collect like terms.

In this lesson

Expand two linear brackets.
Explain where the middle term comes from.

Before you start

Expand a single bracket.
Collect like terms.

01 / LEARN

Understand the method.

THE CORE RULE

(a + b) (c + d) = a c + a d + b c + b d

Four products come from pairing each term in the first bracket with each term in the second.

Account for all four products

Treat each bracket as a whole quantity. Distribute the first term to both terms in the other bracket, then do the same with the second term.

The two middle products are like terms. Add their coefficients while keeping the variable.

\begin{aligned} (x + 3) (x + 5) & = x^{2} + 5 x + 3 x + 15 \\ = x^{2} + 8 x + 15 \end{aligned}

What makes the result quadratic?

A quadratic expression has highest power $2$ in its variable, with a nonzero squared-term coefficient. Its linear or constant term can be zero.

Keep the sign with each term

Read $x - 2$ as $x + (- 2)$ . The negative term must multiply both terms in the other bracket.

Substitute an easy value to check for a mistake. A match at one value is useful evidence, but it does not prove two expressions are always equal.

\begin{aligned} (x - 2) (x + 7) & = x^{2} + 7 x - 2 x - 14 \\ = x^{2} + 5 x - 14 \end{aligned}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Expand, then collect

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

Worked solution

1
Distribute the first bracket.
$(x + 4) (x + 2) = x (x + 2) + 4 (x + 2)$
Both $x$ and $4$ multiply the whole second bracket.
2
Write all four products.
$x^{2} + 2 x + 4 x + 8$
Include the two cross-products as well as the first and last products.
3
Collect the like terms.
$x^{2} + 2 x + 4 x + 8 = x^{2} + 6 x + 8$
Only $2 x$ and $4 x$ are like terms.

Answer

$x^{2} + 6 x + 8$ .

EXAMPLE 02

Include a negative term

Expand $(2 x - 3) (x + 5)$ .

Worked solution

1
Multiply every pair of terms.
$(2 x - 3) (x + 5) = 2 x^{2} + 10 x - 3 x - 15$
The factor $- 3$ produces both $- 3 x$ and $- 15$ .
2
Combine the middle terms.
$2 x^{2} + 10 x - 3 x - 15 = 2 x^{2} + 7 x - 15$
The coefficient of $x$ is $10 - 3 = 7$ .
3
Check the constant term at $x = 0$ .
$(- 3) (5) = - 15$
Both original and expanded forms give $- 15$ . This checks the constant term, not the whole identity.

Answer

$2 x^{2} + 7 x - 15$ .

03 / YOUR TURN

Try it yourself.

Calculate, explain, then apply.

CHECK 01

Try the four products

Expand $(x + 1) (x + 6)$ .

Show answer & reasoning

$x^{2} + 7 x + 6$ .

The products are $x^{2}$ , $6 x$ , $x$ and $6$ . Collect $6 x + x = 7 x$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Expand $(x + 2) (x + 3)$ .
2 marks
Expand $(x - 4) (x + 1)$ .
2 marks
Expand $(x - 5) (x - 2)$ .
2 marks
Expand $(3 x + 2) (x - 1)$ .
2 marks
A rectangle has sides $(x + 2)$ cm and $(x + 7)$ cm. Write its area as an expanded expression.
3 marks
Sam says $(x + 2) (x - 2) = x^{2} - 4 x - 4$ . Correct the expansion and explain the middle term.
3 marks

Read the worked solutions

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

Expand $(x + 2) (x + 3)$ .
2 marks
$x^{2} + 3 x + 2 x + 6 = x^{2} + 5 x + 6$ .
$x^{2} + 5 x + 6$
Expand $(x - 4) (x + 1)$ .
2 marks
$x^{2} + x - 4 x - 4 = x^{2} - 3 x - 4$ .
$x^{2} - 3 x - 4$
Expand $(x - 5) (x - 2)$ .
2 marks
$x^{2} - 2 x - 5 x + 10 = x^{2} - 7 x + 10$ .
$x^{2} - 7 x + 10$
Expand $(3 x + 2) (x - 1)$ .
2 marks
$3 x^{2} - 3 x + 2 x - 2 = 3 x^{2} - x - 2$ .
$3 x^{2} - x - 2$
A rectangle has sides $(x + 2)$ cm and $(x + 7)$ cm. Write its area as an expanded expression.
3 marks
$A = (x + 2) (x + 7) = x^{2} + 7 x + 2 x + 14$ .
$(x^{2} + 9 x + 14) {cm}^{2}$
Sam says $(x + 2) (x - 2) = x^{2} - 4 x - 4$ . Correct the expansion and explain the middle term.
3 marks
$x^{2} - 2 x + 2 x - 4 = x^{2} - 4$ .
The cross-products cancel because $- 2 x + 2 x = 0$ .
$x^{2} - 4$

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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 ↗

In this lesson

Understand the method.

Account for all four products

Keep the sign with each term

See the method in action.

Expand, then collect

Distribute the first bracket.

Write all four products.

Collect the like terms.

Include a negative term

Multiply every pair of terms.

Combine the middle terms.

Check the constant term at x=0.

Try it yourself.

Try the four products

Keep practising

Check the constant term at $x = 0$ .