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Expanding brackets

We multiply the number in front of the bracket by every term inside: 3(x + 4) = 3x + 12. A minus sign in front of the bracket changes the sign of every term: −(x − 5) = −x + 5. Two brackets: every term times every term.

Years 8–9 and upper secondary · Updated

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A number times a bracket

3(x + 4) means three groups of x + 4. That makes three x's and three fours.

3(x + 4) = (x + 4) + (x + 4) + (x + 4)

3(x + 4) = 3x + 12

So the three is multiplied by every term in the bracket.

A minus sign in front of the bracket

A minus sign in front of the bracket means −1 times the bracket. That is why every term changes sign.

−(x − 5) = −1 · x + (−1) · (−5) = −x + 5

Two brackets and squaring a bracket

We want to multiply every term in the first bracket by every term in the second.

(x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6

Two terms times two terms gives four products. Then we collect the x terms. The rules below are shortcuts that give the same answer.

RuleFormulaExample
Squaring a sum(a + b)² = a² + 2ab + b²(x + 5)² = x² + 10x + 25
Squaring a difference(a − b)² = a² − 2ab + b²(x − 4)² = x² − 8x + 16
Difference of two squares(a + b)(a − b) = a² − b²(x + 6)(x − 6) = x² − 36

Example

Example 1: 4(2x − 3) − 2(x + 5)

We want to remove the brackets and then collect like terms.

  1. Four times every term: 4 · 2x = 8x and 4 · (−3) = −12.

    8x − 12 − 2(x + 5)

  2. The minus sign belongs to the two: −2 · x = −2x and −2 · 5 = −10.

    8x − 12 − 2x − 10

  3. We collect like terms: 8x − 2x = 6x and −12 − 10 = −22.

    6x − 22

We put in x = 3. At the start: 4 · 3 − 2 · 8 = −4. In the answer: 18 − 22 = −4. The same value, so it is correct.

Example 2: (x + 3)(x − 2)

We want to multiply every term in the first bracket by every term in the second.

  1. x times both terms: x · x = x² and x · (−2) = −2x.

    x² − 2x

  2. 3 times both terms: 3 · x = 3x and 3 · (−2) = −6.

    x² − 2x + 3x − 6

  3. We collect the x terms: −2x + 3x = x.

    x² + x − 6

We put in x = 4. At the start: 7 · 2 = 14. In the answer: 16 + 4 − 6 = 14. That is correct.

Example 3: squaring a bracket, (2x − 3)²

We want to write (2x − 3)² without brackets. We use the rule for squaring a difference, with a = 2x and b = 3.

  1. a² = (2x)² = 4x².

    4x²

  2. 2ab = 2 · 2x · 3 = 12x, with a minus sign.

    4x² − 12x

  3. b² = 3² = 9, with a plus sign.

    4x² − 12x + 9

We put in x = 3. At the start: (6 − 3)² = 9. In the answer: 36 − 36 + 9 = 9. That is correct.

Pitfalls

  • Changing the sign of only the first term, so that −(x − 5) becomes −x − 5.

    Every term changes sign: −1 · (−5) = +5. The answer is −x + 5.

  • Writing (x + 5)² = x² + 25.

    (x + 5)(x + 5) gives four products: x² + 5x + 5x + 25 = x² + 10x + 25. With x = 1: 6² = 36, not 26.

Practise on your own

0 of 4 correct

  1. Expand: 5(x − 3).

  2. Simplify 12 − (2x − 7).

  3. Multiply out (x + 4)(x − 6).

  4. Multiply out (3x + 2)(3x − 2).

More walkthroughs with Fox

Common questions

What is the difference between expanding and factorising?

Expanding removes the brackets: 3(x + 4) = 3x + 12. Factorising puts them back: 3x + 12 = 3(x + 4). Both have the same value.

Do I have to know the squaring rules by heart?

No, every term times every term always gives the right answer. The rules save time, and we need them when we factorise.

What does conjugate mean?

(x + 6) and (x − 6) have the same terms but different signs in the middle. Brackets like these are called conjugates. Their product is x² − 36.

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