Factorise simple quadratics
Objective. Reverse expansion for monic quadratic expressions.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Factorising a simple quadratic means reversing the expansion process to write x² + bx + c as a product of two brackets. For a monic quadratic (where the coefficient of x² is 1), find two numbers that multiply to give c and add to give b. For example, to factorise x² + 7x + 12, find two numbers multiplying to 12 and adding to 7, which are 3 and 4, giving (x + 3)(x + 4). If c is negative, the two numbers will have opposite signs; if b is negative and c is positive, both numbers are negative. Checking by expanding the brackets afterwards confirms the factorisation is correct.
Worked example
Factorise the quadratic expression x² − 2x − 15.
- 1Find two numbers that multiply to −15 and add to −2.
- 2Testing −5 and 3: −5 × 3 = −15 and −5 + 3 = −2, which works.
- 3Write the factorised form using these numbers: (x − 5)(x + 3).
- 4Check by expanding: x² + 3x − 5x − 15 = x² − 2x − 15.
Answer: (x − 5)(x + 3)
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils often pick two numbers that add correctly but do not multiply to c, such as choosing 2 and 5 for x² + 7x + 12 because they sum to 7 without checking the product is 12. Always verify both the sum and the product before writing the brackets.
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