Expanding Brackets
Single and double brackets, including FOIL.
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 nine products visible.
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.
Expand and simplify (x + 2)(x + 5).
- x² + 10x + 7
- x² + 7x + 7
- x² + 10
- x² + 7x + 10
Show the answer and the working
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.