OxfordAQA IGCSE Mathematics: Algebra Notation and Manipulation — Revision Notes

Condensed for the final weeks. For the full explanation, use the Algebra: Notation and Manipulation study guide.

The eight points (A1–A8)

Almost every point has a Core version and an Extension version — revise them in pairs, since the underlying skill is identical and only the expression’s complexity changes.

Point	Core skill	Extension addition
A2	Transform simple formulae	Subject appears twice
A4	Expand two linear expressions	Expand two/three binomials
A5	Factorise x² + bx + c, difference of two squares	Factorise ax² + bx + c
A6	Index laws, integer powers	Fractional powers
A7	Algebraic fractions, numeric denominators	Linear/quadratic denominators

Worked example: expand then factorise

Expand:      (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6
Factorise:   x^2 + 5x + 6 = (x + 2)(x + 3)

Working an expression both directions checks factorising work — multiply the brackets back out and confirm the original expression returns.

Worked example: algebraic fractions (Extension)

Simplify:  (2x^2 + 3x) / (4x^2 - 9)
Factorise numerator:   x(2x + 3)
Factorise denominator: (2x - 3)(2x + 3)
Cancel shared (2x + 3) factor
Result: x / (2x - 3)

This tests A5 (factorising) and A7 (algebraic fractions) together — the eight points are rarely examined in isolation.

Index laws (A6)

Multiplication: add the powers. Division: subtract the powers. Extension: fractional powers represent roots. Frequently tested through short, calculator-free questions — accuracy without a calculator matters here more than in most sub-topics.

Worked example: transforming a formula (A2)

Rearrange the formula for the area of a circle, A = πr², to make r the subject.

Step 1: divide both sides by pi
        A / pi = r^2
Step 2: take the square root of both sides
        r = sqrt(A / pi)

For the Extension-tier case where the subject appears twice – for example, rearranging a formula where the required variable appears in two separate terms – collect both instances of that variable on one side of the equation first, factorise it out as a common factor, and only then divide to isolate it. This factorise-then-divide sequence is the reliable method whenever a variable cannot be isolated in a single step.

Worked example: factorising ax² + bx + c (Extension)

Factorise 2x² + 5x + 3.

Step 1: find two numbers that multiply to give (a x c) = 2 x 3 = 6,
        and add to give b = 5 -> these numbers are 2 and 3
Step 2: split the middle term using these numbers
        2x^2 + 2x + 3x + 3
Step 3: factorise in pairs
        2x(x + 1) + 3(x + 1)
Step 4: take out the common bracket
        (2x + 3)(x + 1)

This “split the middle term” method extends the Core skill (factorising x² + bx + c, where a = 1 and the two numbers only need to multiply to c) by requiring the numbers to multiply to a×c instead – the same underlying logic, one extra step.

Key terms

Expression — a mathematical phrase with no equals sign (e.g. 3x + 2). Equation — a statement of equality true only for specific values. Identity — a statement of equality true for all values. Factor — a term that divides exactly into another. Common factor — a factor shared by every term in an expression, taken out first when factorising.

Pairing Core and Extension

A student secure on the Core version of A5 (factorising x^2 + bx + c) is most of the way to the Extension version (factorising ax^2 + bx + c) – the extra step is finding factors of a as well as c, not a wholly new method. Revising each point’s Core and Extension forms side by side, rather than as separate topics, is the most efficient route through this sub-topic.

Common mistakes

• Sign errors expanding brackets with a negative term, e.g. (x − 2)(x + 3).
• Forgetting to take a common factor out of every term before factorising a quadratic.
• Applying index laws to terms with different bases as though they were the same.
• Cancelling an additive term in a fraction rather than a genuine multiplicative common factor.
• Treating an identity and an equation as interchangeable.

Quick self-test

1. Expand (x − 2)(x + 3).
2. Factorise x² − 9 (difference of two squares).
3. Simplify x⁵ ÷ x².
4. Why can’t x²y³ and x³y² be combined by adding exponents?
5. What is the key difference between an equation and an identity?
6. Make r the subject of A = pi r^2.

Answers: 1. x² + x − 6. 2. (x − 3)(x + 3). 3. x³. 4. Because the index laws only combine powers of the same base — x²y³ and x³y² have the bases arranged differently, so their exponents cannot simply be added. 5. An equation is true only for specific values of the variable; an identity is true for all values. 6. r = sqrt(A / pi).

How this connects forward

Topic 1 Number establishes arithmetic fluency; this sub-topic is where that fluency generalises into symbolic algebra – the foundation the rest of Topic 2 (functions and graphs, solving equations, sequences) and much of Topics 3 and 4 depend on. A student who is genuinely fluent expanding and factorising here will find solving quadratic equations later in the course considerably faster, since factorising a quadratic is usually the first method taught for solving one.

Official syllabus

OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 — oxfordaqa.com.