Expanding quadratic products | Year 10

01 / LEARN

The idea, step by step

Multiply all four pairs

A quadratic product comes from multiplying two linear expressions. Each term in the first bracket must multiply each term in the second. Keep the signs attached to the terms.

Collect the two middle terms only after the multiplication is complete. A small product table can make a missing pair easier to spot.

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

Recognise a pattern, then check it

Squaring a bracket gives two equal cross-products, not just two squares. Opposite-sign matching brackets make their cross-products cancel.

A pattern is a shortcut, not a different rule. Expand one example directly whenever you are unsure.

{(x + p)}^{2} = x^{2} + 2 p x + p^{2}, (x + p) (x - p) = x^{2} - p^{2}

A common trap

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

Instead: The two cross-products are both $5 x$ , so ${(x + 5)}^{2} = x^{2} + 10 x + 25$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

One positive and one negative constant

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

Each cell records one product.
×	x	−3
x	x²	−3x
4	4x	−12

Multiply to get $x^{2} - 3 x + 4 x - 12$ .
Combine $- 3 x + 4 x = x$ .

x^{2} + x - 12

Example 2

Non-monic brackets

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

The four products are $6 x^{2}$ , $4 x$ , $- 15 x$ and $- 10$ .
The middle terms give $4 x - 15 x = - 11 x$ .

6 x^{2} - 11 x - 10

Check your understanding

Expand $(x + 2) (x + 7)$ .

Show answer

$x^{2} + 9 x + 14$

The middle products are $7 x$ and $2 x$ ; their sum is $9 x$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Expand $(x + 3) (x + 5)$ .
2 marks
Expand $(x - 6) (x + 2)$ .
2 marks
Expand $(2 x + 1) (x - 4)$ .
2 marks
Expand ${(3 x - 2)}^{2}$ .
2 marks
A rectangle has side lengths $(x + 2)$ cm and $(x + 5)$ cm. Write its area as an expanded expression.
3 marks
A student says $(x - 4) (x + 4) = x^{2} + 16$ . Explain the sign error and give the correct expansion.
3 marks

Read the worked solutions

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

Expand $(x + 3) (x + 5)$ .
2 marks
$x^{2} + 5 x + 3 x + 15 = x^{2} + 8 x + 15$ .
$x^{2} + 8 x + 15$
Expand $(x - 6) (x + 2)$ .
2 marks
$x^{2} + 2 x - 6 x - 12 = x^{2} - 4 x - 12$ .
$x^{2} - 4 x - 12$
Expand $(2 x + 1) (x - 4)$ .
2 marks
$2 x^{2} - 8 x + x - 4 = 2 x^{2} - 7 x - 4$ .
$2 x^{2} - 7 x - 4$
Expand ${(3 x - 2)}^{2}$ .
2 marks
Use ${(3 x)}^{2} - 2 (3 x) (2) + 2^{2}$ .
$9 x^{2} - 12 x + 4$
A rectangle has side lengths $(x + 2)$ cm and $(x + 5)$ cm. Write its area as an expanded expression.
3 marks
Area is $(x + 2) (x + 5) = x^{2} + 7 x + 10$ .
The unit is square centimetres.
$(x^{2} + 7 x + 10) {cm}^{2}$
A student says $(x - 4) (x + 4) = x^{2} + 16$ . Explain the sign error and give the correct expansion.
3 marks
The middle terms $4 x$ and $- 4 x$ cancel.
The last product is $(- 4) (4) = - 16$ .
$x^{2} - 16$ ; the constant product is negative.

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Curriculum reference & lesson scope

Australian Curriculum v9.0 and South Australian Curriculum

Year 10 quadratics, including the fuller school-course algebra requested for Ascent: non-monic factorisation, completing the square and the quadratic formula. These lessons consolidate Year 9 monic quadratics and extend the available solving methods.

Official curriculum source ↗

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