Mathematics 4024 · Syllabus tracker
Factorization
Common factors, difference of two squares, quadratics, grouping four terms and solving quadratic equations by factorising.
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Quick revision
Factorization on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Common factors, difference of two squares, quadratics, grouping, and solving by factorising.
1Common factors
- Take out the highest common factor of every term — numbers and letters.
- 'Factorise fully' means nothing more can be taken out.
- Check by expanding back.
Worked example
Factorise fully 12x2 y − 18xy2.
- HCF of 12 and 18 is 6; of x2 y and xy2 is xy.
- 12x2 y ÷ 6xy = 2x and 18xy2 ÷ 6xy = 3y.
6xy(2x − 3y)
2Difference of two squares
- a2 − b2 = (a − b)(a + b). Both terms must be squares and there must be a minus between them.
- 16x2 − 25y2 = (4x − 5y)(4x + 5y).
- Take out a common factor first: 2x2 − 50 = 2(x2 − 25) = 2(x − 5)(x + 5).
- a2 + b2 does not factorise.
Formulas
Worked example
Factorise fully 3x2 − 75.
- Common factor 3: 3(x2 − 25).
- x2 − 25 = (x − 5)(x + 5).
3(x − 5)(x + 5)
3Quadratics x2 + bx + c
- Find two numbers that multiply to c and add to b.
- If c is positive, both numbers have the sign of b. If c is negative, the signs are different.
| Expression | Numbers | Factorised |
|---|---|---|
| x2 + 7x + 12 | 3, 4 | (x + 3)(x + 4) |
| x2 − 7x + 12 | −3, −4 | (x − 3)(x − 4) |
| x2 + x − 12 | 4, −3 | (x + 4)(x − 3) |
| x2 − x − 12 | −4, 3 | (x − 4)(x + 3) |
Worked example
Factorise x2 + 2x − 15.
- Multiply to −15, add to 2: 5 and −3.
(x + 5)(x − 3)
4Quadratics ax2 + bx + c
- Use the ac method: find two numbers that multiply to a × c and add to b.
- Split the middle term with them, then factorise in pairs.
ac method
- Multiply a × c.
- Find two numbers with that product and sum b.
- Rewrite bx as the two terms.
- Factorise the first pair and the second pair.
- Take out the common bracket.
Worked example
Factorise 6x2 + x − 2.
- ac = −12; numbers 4 and −3.
- 6x2 + 4x − 3x − 2.
- 2x(3x + 2) − 1(3x + 2).
(2x − 1)(3x + 2)
5Four terms: grouping
- Group into two pairs, factorise each pair, then take out the common bracket.
- If the brackets don't match, try grouping the terms differently.
Worked example
Factorise xy − 3x + 2y − 6.
- x(y − 3) + 2(y − 3).
(x + 2)(y − 3)
6Solving by factorising
- Rearrange to = 0, factorise, then set each factor equal to 0.
- x(x − 5) = 0 gives x = 0 or x = 5 — never divide both sides by x.
- (2x + 3)(x − 3) = 0 gives x = −3/2 or x = 3.
Worked example
Solve 2x2 − 3x − 9 = 0.
- ac = −18: −6 and 3.
- 2x2 − 6x + 3x − 9 = 2x(x − 3) + 3(x − 3).
- (2x + 3)(x − 3) = 0.
x = 3 or x = −1.5
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| 16x2 − 25 = (4x − 5)2 | That expands to 16x2 − 40x + 25. Use (4x − 5)(4x + 5) |
| 6x2 + 9x = 3(2x2 + 3x) | Take out 3x as well: 3x(2x + 3) |
| (x + 5)(x − 3) = 0 → x = 5 or −3 | Change the signs: x = −5 or 3 |
| Dividing x2 = 5x by x | Rearrange: x(x − 5) = 0, so x = 0 or 5 |
★In the exam
- Always check a factorisation by expanding it in your head.
- If a quadratic will not factorise, the question probably wants the formula.
- 'Hence solve' means use the factorisation you have just found.