Expanding Brackets
Single and double brackets, including FOIL.
Watch the lesson
Every point below, worked through in one video, and each point links straight to its own part.
Learn
Expanding Brackets, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Multiply a single term over a bracket
Multiply EVERYTHING inside by the term outside: 3(2x + 5) = 6x + 15, and −2(x − 4) = −2x + 8 - watch how the minus flips the sign of the second term. Sign slips here are the most common lost mark in algebra.
Expand and simplify expressions with more than one bracket
Expand each bracket separately, then collect like terms: 2(3x + 1) + 3(x − 2) = 6x + 2 + 3x − 6 = 9x − 4. Keep each expansion on its own line until collecting - most errors come from rushing the tidy-up.
Expand double brackets, including squaring a bracket like (x + 3)²
Every term in the first bracket multiplies every term in the second (FOIL): (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6. A squared bracket means the bracket times itself: (x + 3)² = x² + 6x + 9 - never just x² + 9.
Factorise expressions by taking out common factors
Factorising reverses expanding: pull out the highest factor shared by every term. 6x² + 9x = 3x(2x + 3) - take out both the number and the letter. Expand your answer mentally to check it matches what you started with.
Factorise quadratics of the form x² + bx + c, including the difference of two squares
Find two numbers that MULTIPLY to c and ADD to b: for x² + 7x + 12, the pair 3 and 4 gives (x + 3)(x + 4). The difference of two squares has no middle term: x² − 25 = (x + 5)(x − 5) - worth recognising on sight.
Expand products of three binomialsHigher only
Expand two brackets first, then multiply the result by the third. For (x + 1)(x + 2)(x − 3): (x + 1)(x + 2) = x² + 3x + 2, then multiply every term by (x − 3) and collect. Stay organised - a grid keeps all six products visible.
Common mistakes in Expanding Brackets
The slips that cost marks year after year, and what to do instead.
Multiplying only the first term inside the bracket, so 3(2x + 5) becomes 6x + 5.
The number outside multiplies every term inside: 3 × 2x = 6x and 3 × 5 = 15, so 3(2x + 5) = 6x + 15.
Losing a sign with a negative outside the bracket, so −2(x − 4) becomes −2x − 8.
The minus multiplies every term, and a negative times a negative is positive: −2 × x = −2x and −2 × −4 = +8, so −2(x − 4) = −2x + 8.
Squaring each term of a bracket, so (x + 3)² becomes x² + 9.
Write it as (x + 3)(x + 3) and multiply every pair: x² + 3x + 3x + 9 = x² + 6x + 9.
Taking out only part of the common factor, so 6x² + 9x becomes 3(2x² + 3x).
Take out the highest common factor of the numbers and the letters: 6x² + 9x = 3x(2x + 3). Expand your answer to check it.
Picking numbers that add to 7 but do not multiply to 12, so x² + 7x + 12 becomes (x + 6)(x + 1).
The pair must multiply to 12 and add to 7, and only 3 and 4 do both: x² + 7x + 12 = (x + 3)(x + 4).
Practice
Try a Expanding Brackets question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
Answer: x² + 7x + 10
Using FOIL, the middle term comes from 5x + 2x = 7x and the constant from 2 × 5 = 10.
- First: x × x = x².
- Outside: x × 5 = 5x.
- Inside: 2 × x = 2x.
- Last: 2 × 5 = 10.
- Collect: x² + 5x + 2x + 10 = x² + 7x + 10.
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