IGCSE Mathematics: Expanding and Factorising Practice Questions
Expanding means multiplying out brackets, and factorising is the reverse: writing an expression as a product. Every factorisation can be checked instantly by expanding it again.
Topic 2.2 of Cambridge IGCSE Mathematics 0580 supplies the skill that quadratic equations, algebraic fractions and curve sketching all depend on. Factorising a quadratic with a coefficient in front of x squared is the version that appears on Extended papers. The questions below cover both cases.
What you need to know for Expanding and Factorising
- Expanding double bracketsMultiply every term in the first bracket by every term in the second, then collect like terms. Four products result from two binomials.
- Common factorTake out the highest common factor of the numbers and the lowest power of each letter that appears in every term.
- Factorising x<sup>2</sup> + bx + cFind two numbers that multiply to give c and add to give b. Those numbers go into the brackets.
- Factorising ax<sup>2</sup> + bx + cFind two numbers that multiply to ac and add to b, split the middle term using them, then factorise in pairs.
- Difference of two squaresa2 minus b2 factorises to (a + b)(a minus b). Recognise it whenever there are two square terms separated by a minus sign.
- CheckingExpand your factorised answer. If it does not return the original expression, the factorisation is wrong.
IGCSE Mathematics Expanding and Factorising questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Expand and simplify (2x + 3)(x minus 5).
Show the worked answer
- Multiply each term in the first bracket by each term in the second.
- 2x times x = 2x2, and 2x times minus 5 = minus 10x.
- 3 times x = 3x, and 3 times minus 5 = minus 15.
- Collect the like terms: minus 10x + 3x = minus 7x. The expansion is 2x2 minus 7x minus 15.
Factorise fully 12x2y minus 18xy2.
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- Find the highest common factor of the coefficients: the HCF of 12 and 18 is 6.
- Find the lowest power of each letter present in both terms: x appears in both, and y appears in both, so xy comes out.
- The full common factor is 6xy.
- Divide each term by 6xy: 12x2y divided by 6xy = 2x, and 18xy2 divided by 6xy = 3y. So the answer is 6xy(2x minus 3y).
Factorise x2 + 2x minus 15.
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- Look for two numbers that multiply to give minus 15 and add to give plus 2.
- The factor pairs of 15 are 1 and 15, and 3 and 5. Since the product is negative, one number must be negative.
- Plus 5 and minus 3 multiply to minus 15 and add to plus 2.
- So x2 + 2x minus 15 = (x + 5)(x minus 3). Check by expanding: x2 minus 3x + 5x minus 15 = x2 + 2x minus 15.
Factorise 3x2 + 5x minus 2.
Show the worked answer
- Multiply the first and last coefficients: 3 times minus 2 = minus 6. Find two numbers multiplying to minus 6 and adding to 5.
- Those numbers are 6 and minus 1.
- Split the middle term using them: 3x2 + 6x minus x minus 2.
- Factorise in pairs: 3x(x + 2) minus 1(x + 2). The bracket (x + 2) is common, so the answer is (3x minus 1)(x + 2).
Common mistakes in this topic
- Missing products when expanding double brackets.
- Sign errors when one bracket contains a subtraction.
- Not factorising fully, leaving a common factor inside the bracket.
- Failing to spot the difference of two squares.
- Choosing two numbers with the right product but the wrong sum.
Exam tips
- Expand systematically, first term to both, second term to both. Four products every time.
- After factorising, expand your answer mentally to check. It takes seconds and catches almost every error.
- The word fully in a factorising question means look for a common factor before anything else.
- Two square terms with a minus between them are always a difference of two squares.
- For ax squared, use the multiply to ac, add to b method rather than trial and error.
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Expanding and Factorising FAQs
How do I expand double brackets?
Multiply every term in the first bracket by every term in the second, which gives four products from two binomials, then collect the like terms. Working in a fixed order, first term to both then second term to both, ensures no product is missed.
How do I factorise a quadratic like x squared plus 2x minus 15?
Find two numbers that multiply to give the constant term and add to give the coefficient of x. Here 5 and minus 3 multiply to minus 15 and add to 2, so the factorisation is (x + 5)(x minus 3). Check by expanding.
How do I factorise when there is a number in front of x squared?
Multiply the first and last coefficients, then find two numbers that multiply to that product and add to the middle coefficient. Split the middle term using those two numbers and factorise in pairs. The common bracket that emerges gives one factor.
What is the difference of two squares?
An expression of the form a squared minus b squared, which always factorises to (a + b)(a minus b). For example, 9x squared minus 25 factorises to (3x + 5)(3x minus 5). Recognise it by two square terms separated by a subtraction.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.