IGCSE algebra: expansion and factorising is one of the most searched IGCSE maths topics because it appears in almost every paper and underpins quadratics, solving equations, and graphs. Mastering it is essential for top grades in Mathematics 0580 and Additional Maths 0606.

Expansion – What You Need to Know

• Single bracket: (a(b + c) = ab + ac); practise with negative terms and more than two terms inside.
• Double bracket (quadratics): ((x + a)(x + b) = x^2 + (a+b)x + ab); learn the pattern and practise with negatives.
• Difference of two squares: ((a + b)(a - b) = a^2 - b^2) – use it to expand and to factorise.

Always simplify your final answer and check by substituting a value if unsure.

Factorising – Key Types

• Common factor – Take out the highest common factor (e.g. (6x + 12 = 6(x + 2))).
• Quadratics (x^2 + bx + c) – Find two numbers that add to (b) and multiply to (c).
• Difference of two squares – (a^2 - b^2 = (a + b)(a - b)).
• Grouping – For four-term expressions, group in pairs and factorise each pair.

Practise both expansion and factorising in both directions so you can switch between forms.

Worked Example: Expanding Double Brackets

To expand ((x + 3)(x - 2)): multiply First terms ((x \times x = x^2)), Outer ((x \times -2 = -2x)), Inner ((3 \times x = 3x)), Last ((3 \times -2 = -6)). So you get (x^2 - 2x + 3x - 6 = x^2 + x - 6). For factorising (x^2 + x - 6), you need two numbers that add to 1 and multiply to -6: they are 3 and -2, so (x^2 + x - 6 = (x + 3)(x - 2)). Practise several of these until the pattern is automatic.

Where Expansion and Factorising Appear in the Exam

They come up in algebra questions (simplify, solve), in quadratic equations (factorise then use zero product property), and in graphs (expanding to get the form (y = ax^2 + bx + c)). They also link to difference of two squares and completing the square in some syllabuses. So mastering expansion and factorising helps you across several question types.

Common Errors and How to Avoid Them

• Sign errors – With negatives inside brackets, e.g. ((x - 4)(x + 1)), be careful: (-4 \times 1 = -4), so the constant term is -4. Double-check the sign of each term.
• Missing terms – In double brackets, make sure you have all four products (FOIL or equivalent) before simplifying.
• Wrong factorisation – For (x^2 + bx + c), the two numbers must add to (b) and multiply to (c). Check both conditions.
• Not simplifying – After expanding, collect like terms; the answer is usually expected in simplified form.

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