Factorising quadratics, without guessing
Learn how to factorise quadratics using the rectangle of tiles method. Worked examples, exam-style questions with answers.
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Download the PDF (632 KB)Why do two brackets multiply out to x2 + 5x + 6, and how do you get back from one to the other?
Factorising feels like guesswork until you see what it really is: x2 + 5x + 6 is a pile of tiles, and factorising arranges them into one rectangle. The rectangle's two sides are the two brackets.
The picture to keep in your head: A rectangle of tiles
x2 + 5x + 6 is a pile of tiles: one big x-by-x square, five x-by-1 strips and six small 1-by-1 squares. Factorising means arranging all the tiles into one neat rectangle. The rectangle's two side lengths are the two brackets: (x + 2) along one side and (x + 3) along the other. The 6 small squares must fill the corner, so the two numbers multiply to 6; the strips run along the two sides, so the numbers add to 5.
The method, every time
- Take out any common factor first.
- Find two numbers that multiply to the last number and add to the middle number.
- Write them in the brackets: (x + p)(x + q), signs included.
- No middle term? Look for a difference of two squares. A number in front of x2? Split the middle term.
- Expand your brackets to check you get back to the question.
Words you'll see
- factorise
- write as brackets multiplied together
- expand
- multiply out the brackets (the opposite of factorising)
- coefficient
- the number in front of a letter, e.g. 5 in 5x
- difference of two squares
- a square minus a square, e.g. x2 - 49
Watch it explained
- The 10-second trick for factorising quadratics (one-minute video)
- Factorising quadratics with negative numbers (one-minute video)
Factorising is making a rectangle
Factorising is like building a rectangle from a pile of tiles. When you expand (x + 2)(x + 3), you get x² + 5x + 6. Factorising is the opposite: taking the pile of tiles and finding the rectangle’s sides. The two numbers in the brackets must multiply to the last number and add to the middle number. This helps you arrange the tiles into a neat rectangle.
Watch out
- Two numbers that multiply to 6 but don’t add to 5. Always check the numbers add to the middle term
- Forgetting to check by expanding. Expand your answer to make sure it matches the original expression
When the numbers are negative
When you factorise a quadratic with a negative constant term, think of it as a rectangle with some tiles missing. The last number is negative, so one number is positive and one is negative. The middle term’s sign shows which one is bigger. For x² - 2x - 15, you need two numbers that multiply to -15 and add to -2. The bigger number takes the middle term’s sign. This helps you find the right pair of numbers.
Watch out
- Getting the signs the wrong way round, e.g. (x + 5)(x - 3). The bigger number takes the middle term’s sign.
- Treating -15 as 15. Always keep the negative sign with the number.
No middle term? Look for two squares
If there's no middle term, the rectangle has only two squares. One square is big, like x², and the other is small, like 49. To factorise, you need to find two squares that can form a rectangle when one is subtracted from the other. The rule is: x² - a² = (x + a)(x - a).
Watch out
- (x - 7)2. Squares of differences are not the same as difference of squares.
- x2 + 49. Sum of squares doesn't factorise with real numbers.
When there's a number in front of x2
When there's a number in front of x², the tile rectangle gets more squares. The big x² square is now a×x², so you need a×x² tiles. To factorise, find two numbers that multiply to a × c and add to b. Split the middle term and factorise in pairs. This works like arranging more tiles into a rectangle.
Watch out
- Using c instead of a × c. Always multiply a and c first
- Not taking out a common factor first. Factor out any common factor before splitting
- Splitting the middle term incorrectly. Check that the two numbers multiply to a × c and add to b
23 exam-style factorising quadratics questions
Show your working: most marks are for the method. Marks are a guide to how much working an exam would expect. Questions written by Bloom Education in the style of GCSE (AQA, Edexcel, OCR) and Cambridge IGCSE papers.
A · Two positive numbers
- 1Factorise fully: x² + 7x + 12[2 marks]
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(x + 3)(x + 4)
We need two numbers that multiply to 12 and add to 7. These are 3 and 4. So the factorisation is (x + 3)(x + 4). - 2Factorise fully: x² + 8x + 15[2 marks]
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(x + 3)(x + 5)
We need two numbers that multiply to 15 and add to 8. These are 3 and 5. So the factorisation is (x + 3)(x + 5). - 3Factorise fully: x² + 9x + 20[2 marks]
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(x + 4)(x + 5)
We need two numbers that multiply to 20 and add to 9. These are 4 and 5. So the factorisation is (x + 4)(x + 5). - 4Factorise fully: x² + 6x + 8[2 marks]
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(x + 2)(x + 4)
We need two numbers that multiply to 8 and add to 6. These are 2 and 4. So the factorisation is (x + 2)(x + 4).
B · Negative numbers
- 5Factorise fully: 3x² - 6x - 24[2 marks]
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3(x - 4)(x + 2)
Factor out 3: 3(x² - 2x - 8). Find two numbers that multiply to -8 and add to -2: -4 and +2. So (x - 4)(x + 2). Full answer: 3(x - 4)(x + 2) - 6Factorise: x² - 5x + 6[2 marks]
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(x - 2)(x - 3)
Look for two numbers that multiply to +6 and add to -5: -2 and -3. So (x - 2)(x - 3) - 7Factorise: x² + 3x - 10[2 marks]
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(x + 5)(x - 2)
Find two numbers that multiply to -10 and add to +3: +5 and -2. So (x + 5)(x - 2) - 8Factorise: x² - 2x - 15[2 marks]
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(x - 5)(x + 3)
Find two numbers that multiply to -15 and add to -2: -5 and +3. So (x - 5)(x + 3) - 9Factorise: x² - 7x + 12[2 marks]
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(x - 3)(x - 4)
Find two numbers that multiply to +12 and add to -7: -3 and -4. So (x - 3)(x - 4)
C · Difference of two squares
- 10Factorise 4x² - 9[2 marks]
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(2x + 3)(2x - 3)
4x² = (2x)² and 9 = 3², so (2x + 3)(2x - 3). - 11Factorise x² - 81[2 marks]
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(x + 9)(x - 9)
x² - 81 is a difference of two squares. The square root of x² is x. The square root of 81 is 9. So the factors are (x + 9)(x - 9). - 12Factorise fully 2x² - 50[2 marks]
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2(x + 5)(x - 5)
2x² - 50 has a common factor of 2. Factor out 2: 2(x² - 25). x² - 25 is a difference of two squares. So the full factorisation is 2(x + 5)(x - 5). - 13Which of these expressions can be factorised? x² + 16 or x² - 16. Explain why.[2 marks]
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x² - 16 can be factorised because it is a difference of two squares. x² + 16 cannot be factorised.
x² + 16 cannot be factorised because it is not a difference of two squares. x² - 16 is a difference of two squares and can be factorised as (x + 4)(x - 4).
D · Number in front of x^2 (Higher)
- 14Factorise 2x² + 5x + 3[3 marks]
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(2x + 3)(x + 1)
a × c = 6: 2 and 3 add to 5. 2x² + 2x + 3x + 3. - 15Factorise 3x² + 10x + 8[3 marks]
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(3x + 4)(x + 2)
a × c = 24: 6 and 4 add to 10. 3x² + 6x + 4x + 8. - 16Factorise 2x² - x - 6[3 marks]
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(2x + 3)(x - 2)
a × c = -12: -4 and 3 add to -1. 2x² - 4x + 3x - 6. - 17Factorise 5x² - 13x - 6[3 marks]
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(5x + 2)(x - 3)
a × c = -30: -15 and 2 add to -13. 5x² - 15x + 2x - 6. - 18Factorise 6x² + x - 2[3 marks]
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(3x + 2)(2x - 1)
a × c = -12: 4 and -3 add to 1. 6x² + 4x - 3x - 2.
E · Use it
- 19Solve 2x² + 5x - 3 = 0.[3 marks]
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x = 0.5 or x = -3
(2x - 1)(x + 3) = 0, so 2x - 1 = 0 or x + 3 = 0. - 20A rectangle has area x² + 9x + 14. Find expressions for its length and width.[3 marks]
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(x + 2) and (x + 7)
2 × 7 = 14 and 2 + 7 = 9, so x² + 9x + 14 = (x + 2)(x + 7). - 21Work out 101² - 99² without a calculator using difference of two squares.[3 marks]
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400
Use (a² - b²) = (a - b)(a + b). (101 - 99)(101 + 99) = 2 × 200 = 400. - 22Simplify (x² - 9)/(x + 3).[3 marks]
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x - 3
Factorise numerator to (x - 3)(x + 3). Cancel (x + 3). Answer is (x - 3). - 23Solve x² - x - 20 = 0.[3 marks]
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x = 5 or x = -4
(x - 5)(x + 4) = 0, so x = 5 or x = -4.
Factorising quadratics exam checklist
- Look for a common factor first and take it out.
- Last number positive: both signs match the middle term. Negative: one of each.
- Expand your answer to check it.
- No middle term: difference of two squares, (x + a)(x - a).
- A sum of squares like x2 + 16 does not factorise.
- Number in front of x2: find two numbers that multiply to a × c and add to b.
Got stuck on a question?
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All of this page as a 10-page guide, with every answer at the back. No sign-up.
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