Expanding and factorising
Expanding means multiplying out brackets. Factorising is the reverse: putting brackets back in. You need both before you can simplify, solve quadratics or prove things in algebra.
Part 1 of 3: Learn it
In short
- Expanding: multiply every term in one bracket by every term in the other.
- Factorising: always take out the highest common factor first.
- x² + bx + c = (x + p)(x + q) where p × q = c and p + q = b.
Where this is in your specification
Spec points: DfE A4 (expanding three brackets and factorising ax² + bx + c are Higher tier)
| Board | Topic: Quadratics |
|---|---|
| DfE content | A4, A11, A12, A18, A22 |
| AQA 8300 | Algebra |
| Edexcel 1MA1 | Algebra |
| OCR J560 | Algebra |
| Eduqas C300 | HA4, HA11, HA18 |
| Cambridge IGCSE 0580 | E2.2, E2.5, E2.11 |
| National 5 C847 75 | Algebraic skills: completing the square, quadratic functions and equations, the discriminant |
Expanding brackets
- Single bracket: 4(2x − 3) = 8x − 12, and −2(x − 5) = −2x + 10.
- Double brackets: multiply each term by each term (FOIL, or a grid). (x + 6)(x − 2) = x² − 2x + 6x − 12 = x² + 4x − 12.
- A squared bracket means the bracket times itself: (x − 6)² is (x − 6) times (x − 6), which is x² − 12x + 36.
Factorising into one bracket
Find the highest common factor of every term, numbers and letters, and write it outside. 12x² − 18x = 6x(2x − 3). Expand your answer to check it.
Factorising quadratics
For x² + bx + c, find two numbers that multiply to c and add to b. For x² − 3x − 28, the pair is −7 and +4, so it is (x − 7)(x + 4).
- Difference of two squares: a² − b² = (a − b)(a + b). For example 9y² − 25 = (3y − 5)(3y + 5).
- Higher: for ax² + bx + c, find two numbers that multiply to ac and add to b, split the middle term, and factorise in pairs.
Expand (x − 9)(x + 2).
Show the answer
x² + 2x − 9x − 18 = x² − 7x − 18.
Part 2 of 3: See it worked
Worked examples
Example 1
Expand and simplify (2x + 3)(x − 5).
- 2x × x = 2x²
- 2x × (−5) = −10x and 3 × x = 3x
- 3 × (−5) = −15
- 2x² − 10x + 3x − 15
Answer: 2x² − 7x − 15.
Example 2
Factorise 3x² + 10x + 8. (Higher)
- a × c = 3 × 8 = 24; two numbers that multiply to 24 and add to 10: 6 and 4
- Split the middle: 3x² + 6x + 4x + 8
- Factorise in pairs: 3x(x + 2) + 4(x + 2)
- = (3x + 4)(x + 2)
Answer: (3x + 4)(x + 2).
Common mistakes
- Forgetting the middle term: (x + 3)² is x² + 6x + 9, not x² + 9.
- Multiplying only the first term inside a bracket: 3(x + 4) is 3x + 12.
- Sign slips with a negative outside: −(x − 2) is −x + 2.
- Not taking out the whole common factor: 8x² + 12x = 4x(2x + 3), not 2x(4x + 6).
Factorise x² + 11x + 24.
Show the answer
3 × 8 = 24 and 3 + 8 = 11, so (x + 3)(x + 8).
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Expand (x − 9)(x + 2).
x² + 2x − 9x − 18 = x² − 7x − 18.
Factorise x² + 11x + 24.
3 × 8 = 24 and 3 + 8 = 11, so (x + 3)(x + 8).
Factorise 4x² − 49.
Difference of two squares: (2x − 7)(2x + 7).
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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