Teach Me Maths

Expanding Single Brackets

On this page

How to multiply out a single bracket by multiplying every term inside it, how to deal with minus signs and letters outside the bracket, and how to expand and simplify two single brackets added or subtracted.

What expanding a bracket means

To expand a bracket means to multiply it out, so that the answer has no brackets. Multiply every term inside the bracket by the term outside.

For example, 3(x+4) means three lots of x+4. That is three lots of x and three lots of 4:

3(x+4)=3x+12

You can picture it as a rectangle 3 high and x+4 wide. Split the width into x and 4. The two smaller areas are 3x and 12, so the whole area is 3x+12

In the exam you are usually asked to expand one bracket, such as 6(2n-1), for 1 mark. A question to expand and simplify two single brackets, such as 2(x+3)+4(x-1), is usually worth 2 marks. Expanding is also the first step in solving many equations.

Worked example 1: Expanding a bracket

Expand 4(x+3)

  1. Multiply the 4 by the first term: 4\times x=4x

  2. Your turn. Now multiply the 4 by the second term. Work out 4\times3

    4\times3=12

  3. Together that is 4x+12. Check with x=2: 4\times(2+3)=20 and 4\times2+12=20
  4. Answer: 4x+12

Minus signs inside the bracket

Each term inside a bracket keeps the sign in front of it. In 5(x-2) the two terms are x and -2

When you multiply a positive number by a negative number, the answer is negative. So 5\times(-2)=-10 and 5(x-2)=5x-10

Worked example 2: A minus sign inside

Expand 4(3y-2)

  1. The terms inside are 3y and -2. Both are multiplied by the 4 outside.
  2. 4\times3y=12y

  3. Your turn. What is 4\times(-2)?

    4\times(-2)=-8

  4. Answer: 12y-8

When the term outside has a letter

Multiply the numbers, then the letters, and write the number first. For example, 3\times2x=6x and 2x\times4=8x

When a letter is multiplied by itself, the answer is that letter squared: x\times x=x^2. When you multiply powers of the same letter, add the powers, so x\times x^2=x^3

So x(x+7)=x^2+7x. The x^2 term and the 7x term are different kinds of term, so they stay separate.

Worked example 3: A letter outside the bracket

Expand 5t(t-3)

  1. The term outside is the whole of 5t. Multiply it by t and by -3

  2. Your turn. What is 5t\times t?

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    5t\times t=5t^2

  3. 5t\times(-3)=-15t
  4. The terms 5t^2 and -15t are not like terms, so they cannot be collected.
  5. Answer: 5t^2-15t

Expand and simplify

When a question says expand and simplify, there are two jobs. First multiply out every bracket. Then collect like terms: put the x terms together and the number terms together.

Each term takes the sign in front of it with it. In 4x+9-x-2 the x terms make 3x and the numbers make 7, so the answer is 3x+7

Worked example 4: Adding two brackets

Expand and simplify 2(4x+3)+5(x-1)

  1. Expand the first bracket: 2(4x+3)=8x+6
  2. Expand the second bracket: 5(x-1)=5x-5
  3. Write the four terms together: 8x+6+5x-5

  4. Your turn. Collect the x terms. What is 8x+5x?

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    8x+5x=13x

  5. Collect the numbers: 6-5=1
  6. Answer: 13x+1

Subtracting a bracket

When a bracket is subtracted, the minus sign belongs to the number in front. In -3(x-2) the term outside is -3, not 3.

Negative times positive is negative, and negative times negative is positive. So -3\times x=-3x and -3\times(-2)=6, which gives -3(x-2)=-3x+6

When there is just a minus sign in front of a bracket, it means multiply by -1. So -(k-6)=-k+6

Worked example 5: Subtracting a bracket

Expand and simplify 6(x+1)-2(x-4)

  1. Expand the first bracket: 6(x+1)=6x+6
  2. The second bracket is multiplied by -2, so -2\times x=-2x

  3. Your turn. What is -2\times(-4)?

    -2\times(-4)=+8

  4. So the whole expression is 6x+6-2x+8
  5. Collect like terms: 6x-2x=4x and 6+8=14
  6. Answer: 4x+14

Worked example 6: A perimeter with brackets

A rectangle is (2x+5) cm long and (8-x) cm wide. Find an expression for the perimeter of the rectangle. Give your answer in its simplest form.

  1. The perimeter is two lengths and two widths: 2(2x+5)+2(8-x)
  2. Expand the first bracket: 2(2x+5)=4x+10

  3. Your turn. Expand 2(8-x)

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    2(8-x)=16-2x

  4. Write all the terms together: 4x+10+16-2x
  5. Collect like terms: 4x-2x=2x and 10+16=26
  6. Answer: the perimeter is (2x+26) cm

Common mistakes

  • Multiplying only the first term. 4(2n+5) is 8n+20, because the 4 multiplies the 5 as well.
  • Adding instead of multiplying. 3(x+4) is 3x+12: the 3 multiplies each term and is never added on.
  • Losing a minus sign. Take the sign in front of each number with it: -2(x-4) is -2x+8, as minus times minus is plus.
  • Getting muddled when the letter comes second. 3(5-2x) is 15-6x, because the minus sign belongs to the 2x.
  • Collecting terms that are not alike. Only put x terms with x terms and numbers with numbers, so 5x+3 stays as it is.
  • Getting the powers wrong. x\times x is x^2, not 2x, and x\times x^2 is x^3, so add the powers.

Practice questions

More on this topic: Expanding Single Brackets worksheet with full solutions