Why Brackets Matter So Much at GCSE
Expanding and factorising are the two directions of the same journey. Expanding removes brackets: becomes . Factorising puts them back: becomes . Between them they underpin a huge slice of the algebra on every GCSE paper, because solving equations, simplifying expressions, working with quadratics and proving algebraic statements all lean on these two skills.
The questions themselves are usually short, worth 1 to 3 marks each, but the skill is embedded inside longer questions too. A 5-mark simultaneous equations problem or a quadratic to be solved by factorising both fall apart if the bracket work goes wrong on line one. That is why examiners' reports mention sign errors in expansion year after year: a single slipped minus sign early in a question can quietly cost every mark after it.
This guide works through each technique in exam order of difficulty: single brackets, double brackets, factorising with a common factor, factorising quadratics, the difference of two squares, and finally the Higher-tier case where the coefficient is bigger than 1.
- Expanding removes brackets; factorising introduces them: they are inverse operations
- Short questions are worth 1 to 3 marks, but the skill hides inside longer questions too
- Sign errors are the single biggest source of lost marks on this topic
Expanding Single Brackets
To expand a single bracket, multiply the term outside by every term inside. This is the distributive law, and the word "every" carries all the danger: with , both the and the must be multiplied by 3.
Negative numbers outside the bracket are where marks leak. Expanding gives : the times makes positive 8. Whenever the outside term is negative, expect both signs inside to flip, and pause to check they have.
Expressions such as combine both ideas: expand each bracket separately, keeping the signs honest, then collect like terms. Here that gives . Notice the terms vanished entirely; exam questions sometimes engineer this deliberately, and reaching a clean number is often the sign you have expanded correctly.
Expand and simplify
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Expand the first bracket: and , giving
- 2
Expand the second bracket carefully: and
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Write the whole expression:
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Collect like terms: and
Expanding Double Brackets (FOIL)
To expand two brackets multiplied together, every term in the first bracket must multiply every term in the second. With two terms in each bracket that means four multiplications, and FOIL is the standard way to keep track: First, Outer, Inner, Last. After the four products, collect the like terms, which are almost always the two middle terms.
FOIL is not the only layout. Many students prefer a 2-by-2 grid, writing one bracket across the top and one down the side, then filling in the four cells. The grid makes it much harder to miss a term, which is the main failure mode, so if your FOIL expansions keep coming out with three terms before simplifying instead of four, switch to the grid.
When the brackets contain coefficients, the same four multiplications apply. Expanding gives . Nothing new is happening; there is just more arithmetic riding on each step.
Expand
- 1
First:
- 2
Outer:
- 3
Inner:
- 4
Last:
- 5
Collect the middle terms:
The Squared Bracket Trap
Here is the most reliably marked-wrong line in GCSE algebra: . It looks plausible and it is wrong. A squared bracket means the bracket multiplied by itself, so , and expanding properly gives . The middle term is not optional.
The general pattern is worth knowing by heart: , and . Note that the final term is positive in both, because a negative times a negative is positive.
This expansion is exactly what completing the square runs on, so getting fluent here pays off directly in solving quadratic equations. Whenever you see a squared bracket, write it out as two brackets at least until the pattern is automatic; the ten extra seconds are cheaper than the lost marks.
- $(x + 5)^2$ means $(x + 5)(x + 5)$, never $x^2 + 25$
- Pattern: $(x + a)^2 = x^2 + 2ax + a^2$
- The constant term is positive even when the bracket contains a minus
- Write the bracket twice and expand until the pattern is second nature
Factorising with a Common Factor
Factorising into a single bracket reverses single-bracket expansion. Find the highest common factor (HCF) of all the terms, including any letters, write it outside the bracket, and fill the bracket with what remains after dividing each term by it.
The examiner's word to watch is "fully". Factorising as or is not wrong, but it is not fully factorised, because the terms inside still share a factor, and the question will usually be marked accordingly. The complete answer is : the HCF of 12 and 8 is 4, and both terms contain at least one .
Always check your answer by expanding it back in your head. and : the original expression returns, so the factorisation is right. This thirty-second habit converts "I think it's right" into "I know it's right" on every factorising question you ever answer.
Factorise fully
- 1
Find the HCF of the numbers: HCF of 12 and 8 is 4
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Both terms contain , so the full HCF is
- 3
Divide each term by : and
- 4
Write the factorised form and check by expanding: ✓
Factorising Quadratics ($x^2 + bx + c$)
For a quadratic in the form , the task is to find two numbers that multiply to give and add to give . Those two numbers drop straight into the brackets: , because and .
The signs of and tell you what kind of numbers to hunt for, and reading them first saves a lot of trial and error. If is positive, both numbers have the same sign, and tells you which sign it is. If is negative, the numbers have opposite signs, and tells you which of them is bigger.
So for : the product is , so one number is positive and one negative; the sum is , so the negative one wins. The pair is and , giving . Expand to confirm: ✓.
Factorise
- 1
You need two numbers that multiply to and add to
- 2
is negative, so the two numbers have opposite signs; is negative, so the larger one is negative
- 3
Try the factor pairs of 15: and give and ✓
- 4
Write the brackets:
- 5
Check by expanding: ✓
Difference of Two Squares
One factorising pattern deserves its own name because it looks unfactorisable at first glance: . An expression like has no middle term and a negative constant, which rules out the usual two-numbers approach, yet it factorises perfectly: . Expand it and the middle terms and cancel, which is why no term appears.
Spotting the pattern is the whole skill. Ask two questions: are both terms perfect squares, and is there a minus between them? passes both tests, since and , so it factorises as . But fails (a sum of squares does not factorise at GCSE), and so does , since 20 is not a square number.
The pattern also powers a lovely non-calculator trick: , no long multiplication needed. Questions of exactly that shape appear on non-calculator papers, dressed up as "work out" questions that reward the student who recognises the structure.
- $a^2 - b^2 = (a + b)(a - b)$: no middle term, minus in the middle
- $x^2 - 25 = (x + 5)(x - 5)$ and $4x^2 - 9 = (2x + 3)(2x - 3)$
- A sum of two squares, like $x^2 + 25$, does not factorise at GCSE
- Use it numerically: $101^2 - 99^2 = 200 \times 2 = 400$
Higher Tier: When the $x^2$ Coefficient Isn't 1
Higher papers include quadratics like , where the number in front of is bigger than 1. The reliable approach is the ac method. Multiply and : here . Find two numbers that multiply to that product and add to : for a product of 6 and a sum of 7, the numbers are 6 and 1. Use them to split the middle term, then factorise in pairs.
Splitting gives . Factorise the first pair: . Factorise the second pair: . Both halves now contain the bracket , which is your check that the split was right, and pulling it out gives .
This is the version of factorising that feeds directly into solving harder quadratics, where each bracket is set equal to zero. If factorising fails or gets slow, the quadratic formula always works; knowing when to switch methods is covered in the quadratics topic guide.
Factorise
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Multiply :
- 2
Find two numbers that multiply to 6 and add to 7: they are 6 and 1
- 3
Split the middle term:
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Factorise each pair:
- 5
Both pairs share , so pull it out:
- 6
Check: ✓
The Sign Errors That Cost the Most Marks
Nearly all lost marks on this topic are sign slips, and they cluster in predictable places. A negative outside a bracket flips every sign inside, so is , not ; check the second term especially, because that is the one students forget to flip. In double brackets, the Last multiplication with two negatives comes out positive: ends in , not .
Two more habits protect you. First, never skip the collecting step: write all four terms of an expansion before simplifying, because doing it in your head is where middle terms go missing. Second, after any factorisation, spend a few seconds expanding your answer back. It is the one check that catches every error type at once, and it costs almost nothing.
Exam Ladder generates fresh practice on this exact skill from a bank of over 280,000 practice questions across 70 GCSE topics, and its worked solutions show the sign of every term at every step, which is precisely where the feedback needs to be. A wider tour of high-frequency slips across all topics is in the most common GCSE maths mistakes.
- A minus outside a bracket flips every sign inside: $-(x - 3) = -x + 3$
- Negative times negative is positive: $(x - 2)(x - 5)$ ends in $+10$
- Write all four terms of an expansion before collecting
- Always check a factorisation by expanding it back
Frequently asked questions
What's the difference between expanding and factorising?
Expanding removes brackets, for example 3(x + 2) becomes 3x + 6. Factorising introduces brackets, so 3x + 6 becomes 3(x + 2). They are inverse operations, which is why you can always check a factorisation by expanding it.
How do I factorise if the coefficient of x² isn't 1?
Use the ac method: for ax² + bx + c, multiply a by c, find two numbers that multiply to that product and add to b, split the middle term with them, then factorise in pairs. This is Higher tier content.
Why is (x + 5)² not equal to x² + 25?
Because squaring a bracket means multiplying it by itself. (x + 5)(x + 5) produces two middle terms, 5x and 5x, so the correct expansion is x² + 10x + 25. Forgetting the middle term is one of the most common errors examiners report.
What does "factorise fully" mean?
It means no further common factor should remain inside the bracket. Writing 12x² + 8x as 2(6x² + 4x) is incomplete because 6x² and 4x still share a factor of 2x; the full answer is 4x(3x + 2).
How do I spot a difference of two squares?
Look for exactly two terms, both perfect squares, with a minus between them and no middle term. Then a² − b² factorises as (a + b)(a − b). A sum of two squares, like x² + 25, does not factorise at GCSE.