Expanding brackets

If you are asked to work out the value of 3(2 + 4), the usual method would be to evaluate the bracket first and multiply the answer by 3. Performing this calculation would give you 18.

You could instead multiply the number outside the bracket by each of the two numbers inside the bracket, and add the result together. This would give you 3 × 2 + 3 × 4, which also gives the result 18.

This strategy is particularly important when we have to work out the value of a bracket containing letters such as 3(\({a}\) + 2), because it is impossible to evaluate the bracket first. We cannot simplify \({a}\) + 2.

Using the second method, however, would leave us with 3 × \({a}\) + 3 × 2 which we can write as 3\({a}\) + 6. When you are asked to expand/multiply out an algebraic expression with a bracket, this is the method you must employ.

Example

Expand 2(3\({y}\) - 4)

Solution

The rule is that the number outside the bracket must be multiplied by both numbers inside the bracket and then the results must be added together. This leaves us with (2 × 3\({y}\)) + (2 × -4) = 6\({y}\) – 8.

Question

Expand 6(2\({z}\) + 2)

Some of the more difficult questions you will be asked will involve expanding and simplifying an expression including two brackets, such as:

3(\({p}\) + 2) + 2(\({p}\) - 4)

The method used to solve this is exactly the same as before. We treat each bracket separately, then simplify our answer as far as possible.

For the first bracket: 3(\({p}\) + 2) = 3\({p}\) + 6

For the second bracket: 2(\({p}\) – 4) = 2\({p}\) – 8

Grouping like terms gives: 3\({p}\) + 2\({p}\) + 6 – 8

Simplifying leaves us with our final answer: 5\({p}\) – 2

Example

Simplify 3(\({i}\) + 4) – 2(\({i}\) –3)

Solution

We use exactly the same method as before but we must be especially careful with the signs, remembering that the sign belongs to the number that follows it.

The first bracket is pretty straightforward: 3(\({i}\) + 4) = 3\({i}\) + 12

The second bracket gives us: –2(\({i}\) – 3) = (–2 × \({i}\)) + (–2 × -3) = –2\({i}\) + 6

Grouping like terms: 3\({i}\) – 2\({i}\) + 12 + 6

Simplifying: \({i}\) + 18

Question

Simplify –3(\({k}\) – 2) + –2(\({k}\) + 4)

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Basic algebra – WJECExpanding brackets