To expand single brackets, multiply the term outside the bracket by every term inside it, then write the results as a sum. This uses the distributive law. For example, 3(x + 4) becomes 3 × x + 3 × 4 = 3x + 12. The key is to miss no term and to handle signs carefully.
What does "expand the brackets" mean?
Expanding (or "multiplying out") brackets means rewriting an expression without the brackets, by multiplying the outside term by each inside term. It is the reverse of factorising. The rule that lets you do it is the distributive law: a(b + c) = ab + ac.
How do you expand a single bracket step by step?
Follow the same routine every time:
- Identify the term outside the bracket.
- Multiply it by the first term inside.
- Multiply it by the second term inside.
- Write both products as a sum, keeping the signs.
For 5(2x + 3): 5 × 2x = 10x and 5 × 3 = 15, giving 10x + 15.
How do you handle negative signs?
Signs are where most marks are lost. A negative outside the bracket flips the sign of every term inside.
| Expression | Working | Answer |
|---|---|---|
2(x + 5) |
2x + 10 |
2x + 10 |
4(x − 3) |
4x − 12 |
4x − 12 |
−3(x + 2) |
−3x − 6 |
−3x − 6 |
−2(x − 4) |
−2x + 8 |
−2x + 8 |
Notice the last row: a negative times a negative gives a positive (−2 × −4 = +8).
Worked example with a variable outside
Expand x(x + 7).
x × x = x²(multiplying a variable by itself gives a power)x × 7 = 7x- Answer:
x² + 7x
This shows that the outside term can be a variable, not just a number.
What about expanding and then simplifying?
Often you expand and then collect like terms. Expand 3(x + 2) + 4:
- Expand:
3x + 6 + 4 - Collect the constants:
6 + 4 = 10 - Answer:
3x + 10
Always check whether the final expression can be simplified further.
What are the most common mistakes?
The classic errors are forgetting to multiply the second term (writing 3(x + 4) = 3x + 4) and mishandling a negative outside the bracket. Slow down, multiply every term, and double-check each sign before moving on.
How is expanding brackets the opposite of factorising?
Expanding and factorising are mirror images of each other. Expanding takes 3(x + 4) and turns it into 3x + 12. Factorising does the reverse: it takes 3x + 12, spots the common factor of 3, and rewrites it as 3(x + 4). Seeing the two as a pair helps enormously, because you can always check a factorisation by expanding it again — if you get back to where you started, you factorised correctly. At KS3 you meet expanding first, then factorising in Year 8, and the two skills are used together constantly when simplifying expressions and, later, solving quadratic equations at GCSE. Treat them as one connected idea rather than two separate rules.
Frequently asked questions
How do you expand single brackets?
Multiply the term outside the bracket by every term inside it, then add the results. For example, 3(x + 4) becomes 3 × x + 3 × 4, which is 3x + 12. This is an application of the distributive law.
What is the distributive law?
The distributive law states that a(b + c) = ab + ac. It means multiplying a single term across a sum gives the same result as multiplying each part separately and adding them. It is the rule that justifies expanding brackets.
How do you expand a bracket with a negative outside it?
Multiply each inside term by the negative, which flips its sign. For example, −3(x + 2) becomes −3x − 6, and −2(x − 4) becomes −2x + 8, because a negative times a negative gives a positive.
What is the most common mistake when expanding brackets?
The most common mistake is multiplying only the first term inside the bracket and forgetting the rest, such as writing 3(x + 4) = 3x + 4. The second most common is mishandling a negative sign outside the bracket.
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