Factorising quadratics, explained with examples
Factorising a quadratic means writing it as a product of two brackets. It is the quickest way to solve many quadratic equations and appears on every GCSE Maths paper.
On this page
- What factorising means
- Step 0: always look for a common factor
- Type 1: x² + bx + c
- Type 2: the difference of two squares
- Type 3: ax² + bx + c with a ≠ 1 (Higher)
- Solving equations by factorising
- Checking your answer
- Factorising in algebraic fractions
- When factorising does not work
- Common mistakes
- Practice questions
- Frequently asked questions
- Key takeaways
- Related guides
What factorising means
Expanding turns brackets into an expression: (x + 2)(x + 3) = x² + 5x + 6. Factorising does the reverse: it starts with x² + 5x + 6 and finds (x + 2)(x + 3). Factorising is useful because if a product equals zero, one of the factors must be zero. That turns one hard equation into two easy ones.
Step 0: always look for a common factor
Before anything else, check whether every term shares a factor. Taking it out first makes the rest much easier.
Example 1: a common factor
Factorise x² + 7x and 3x² − 12x.
- Both terms of x² + 7x contain x: x(x + 7).
- Both terms of 3x² − 12x contain 3x: 3x(x − 4).
Type 1: x² + bx + c
Find two numbers that multiply to c and add to b. Those numbers go in the brackets: (x + p)(x + q). Use the signs to guide your search:
- c positive and b positive: both numbers positive.
- c positive and b negative: both numbers negative.
- c negative: one number positive and one negative; the bigger one has the same sign as b.
Example 2: both positive
Factorise x² + 7x + 12.
- Pairs that multiply to 12: 1 and 12, 2 and 6, 3 and 4. Only 3 and 4 add to 7.
- (x + 3)(x + 4).
Example 3: both negative
Factorise x² − 9x + 20.
- Multiply to +20, add to −9: −4 and −5.
- (x − 4)(x − 5).
Example 4: a negative constant
Factorise x² + 2x − 15.
- Multiply to −15, add to +2: +5 and −3.
- (x + 5)(x − 3).
Type 2: the difference of two squares
When there is no x term and the expression is one square minus another, use a² − b² = (a + b)(a − b). Look out for this pattern, because it is quick to spot and often appears in non-calculator questions.
Example 5: difference of two squares
Factorise x² − 49 and 4x² − 25.
- x² − 49 = (x + 7)(x − 7).
- 4x² − 25 = (2x)² − 5² = (2x + 5)(2x − 5).
Type 3: ax² + bx + c with a ≠ 1 (Higher)
The ac method (also called splitting the middle term) is reliable:
- Multiply a by c.
- Find two numbers that multiply to ac and add to b.
- Split the middle term bx using those two numbers.
- Factorise the first pair and the second pair separately, then take out the common bracket.
Example 6: the ac method
Factorise 2x² + 7x + 3.
- ac = 6. Numbers that multiply to 6 and add to 7: 6 and 1.
- Split: 2x² + 6x + x + 3.
- Group: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Example 7: with negatives
Factorise 3x² − 10x − 8.
- ac = −24. Multiply to −24 and add to −10: −12 and 2.
- Split: 3x² − 12x + 2x − 8 = 3x(x − 4) + 2(x − 4).
- = (3x + 2)(x − 4).
Example 8: common factor first
Factorise 2x² + 10x + 12 fully.
- Take out 2: 2(x² + 5x + 6).
- = 2(x + 2)(x + 3). “Fully” in the question is a signal to look for a common factor.
Solving equations by factorising
To solve a quadratic equation, rearrange it to equal zero, factorise, then set each bracket equal to zero.
Example 9: solving an equation
Solve x² − x − 12 = 0.
- Factorise: (x − 4)(x + 3) = 0.
- So x − 4 = 0 or x + 3 = 0, giving x = 4 or x = −3.
Example 10: rearranging first
Solve 2x² = 5x + 3.
- Rearrange: 2x² − 5x − 3 = 0. ac = −6; use −6 and 1.
- 2x² − 6x + x − 3 = 2x(x − 3) + 1(x − 3) = (2x + 1)(x − 3).
- x = −1/2 or x = 3.
Checking your answer
Always expand your brackets to check. If (x + 5)(x − 3) expands back to x² + 2x − 15, you are right. When solving an equation, substitute each answer into the original equation. This takes a few seconds and catches sign errors, which are the most common mistake in factorising.
Factorising in algebraic fractions
Factorising is also the key to simplifying algebraic fractions, a common Higher tier question. Factorise the top and the bottom, then cancel any bracket that appears in both. For example, (x² − 9)/(x² + 5x + 6) = (x + 3)(x − 3)/((x + 3)(x + 2)) = (x − 3)/(x + 2). You can only cancel whole brackets, never individual terms.
When factorising does not work
Not every quadratic factorises with whole numbers. If you cannot find two numbers after a sensible search, or if the question asks for answers to decimal places, use the quadratic formula or completing the square instead. A quick test: if b² − 4ac is a perfect square, the quadratic factorises.
Common mistakes
- Forgetting to take out a common factor first.
- Choosing numbers that multiply to c but do not add to b, or the other way round.
- Sign errors when c is negative.
- Solving (x − 4)(x + 3) = 0 as x = −4 or x = 3; the signs flip when you solve.
- Trying to factorise before rearranging the equation to equal zero.
Practice questions
Try each one before opening the answer.
Q1. Factorise x² + 11x + 24.
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Q2. Factorise x² − 3x − 28.
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Q3. Factorise 9x² − 16.
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Q4. Factorise 5x² + 13x − 6.
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Q5. Solve x² + 6x + 8 = 0.
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Frequently asked questions
Is factorising quadratics on the Foundation paper?
Yes, for x² + bx + c and the difference of two squares. Quadratics where a is not 1 are on the Higher tier.
Which is the quickest method for hard quadratics?
With practice, many students can spot factor pairs directly. The ac method is the most reliable when you cannot, and the quadratic formula is the fallback when nothing factorises.
Key takeaways
- Always take out common factors first.
- For x² + bx + c: two numbers that multiply to c and add to b.
- a² − b² = (a + b)(a − b).
- For ax² + bx + c: use the ac method.
- Check by expanding; solve by setting each bracket to zero.