Algebra for OxfordAQA IGCSE Mathematics: Methods and Worked Examples

Algebra is where the marks are won or lost

Here is the blunt version: if you are weak at oxfordaqa igcse mathematics algebra, you will not clear a good grade, full stop. Algebra is not one isolated topic on the paper, it is the language the rest of the paper is written in. A geometry question that asks you to "find x" is an algebra question wearing a triangle costume. A statistics question asking you to find a formula for the nth term of a pattern is an algebra question wearing a spreadsheet costume. Sort algebra out and half the paper gets easier by association.

This igcse 9260 algebra guide covers the four named topics on the specification: notation and manipulation, functions/graphs/calculus, solving equations and inequalities, and sequences. Work through it in order, do every worked example by hand, and don't skip to the self-check questions at the end without earning them first. Consider this your oxfordaqa igcse mathematics explained resource for algebra specifically: no padding, just the methods, the traps, and the practice you need.

Notation and manipulation

This is the toolkit topic. Everything else in algebra depends on you being fast and accurate here.

Collecting like terms and expanding brackets

Collecting like terms means grouping terms with the same letter and power. Expanding a single bracket means multiplying everything inside by whatever sits outside. Expanding two brackets multiplied together means every term in the first multiplies every term in the second.

Factorising

Factorising reverses expansion. Start by taking out any common factor, then check whether what remains fits a quadratic pattern, including the difference of two squares (a² - b² = (a + b)(a - b)) or a trinomial x² + bx + c. At Extension you also need ax² + bx + c, where the leading coefficient is not 1.

Index laws and algebraic fractions

The same index laws you use in Number apply to algebra: a^m × aⁿ = a^m+n, a^m ÷ aⁿ = a^m-n, and at Extension, fractional powers. You are also expected to add, subtract, multiply and divide algebraic fractions, with numeric denominators at Core and linear or quadratic denominators at Extension.

Solving equations and inequalities

An equation question rewards a clean, methodical layout as much as it rewards the right final answer, because method marks are awarded line by line.

• Linear equations in one unknown, including brackets and the unknown appearing on both sides.
• Quadratic equations solved by factorising, and at Extension, by completing the square or the quadratic formula.
• Simultaneous equations in two variables, solved algebraically, and at Extension, one linear and one quadratic.
• Linear inequalities in one variable, represented on a number line, and at Extension, quadratic inequalities and inequalities on a graph.

For simultaneous equations, the elimination method is usually faster than substitution when both equations are already in a similar form. Multiply one or both equations so that the coefficient of one variable matches, then add or subtract to eliminate it.

For inequalities, remember the one rule that trips up almost everyone: multiplying or dividing both sides by a negative number reverses the inequality sign. On a number line, an open circle marks a strict inequality (< or >) and a closed (filled) circle marks an inequality that includes the boundary (≤ or ≥). On a graph, a dashed line means the boundary itself is not included, and a solid line means it is.

Functions, graphs and calculus

This topic connects algebra to pictures, and OxfordAQA rewards being able to move between an equation and its graph in either direction.

Straight lines

The form y = mx + c is central: m is the gradient, c is the y-intercept. Parallel lines share the same gradient. At Extension, perpendicular lines have gradients that multiply to -1, and you should be able to find the equation of a line through two given points or through one point with a known gradient.

Functions and function notation

At Core level, you interpret simple expressions as functions with inputs and outputs. At Extension, this becomes formal function notation: f(x) = ..., together with domain and range, and the composite function fg (apply g first, then f) and the inverse function f^-1. A common slip is applying a composite function in the wrong order; fg(x) means g acts on x first, and the result then feeds into f.

Quadratics and other curves

You should recognise and sketch linear, quadratic, simple cubic and reciprocal graphs (y = 1/x), and at Extension, exponential and trigonometric graphs. For a quadratic, identify the roots (where it crosses the x-axis), the y-intercept, and the turning point, which at Extension you find algebraically by completing the square.

Calculus, Extension only

Differentiation at this level is limited to kxⁿ terms, where n is a positive integer or 0. The gradient function dy/dx tells you the gradient of the curve at any point. Setting dy/dx = 0 finds stationary points: maxima, minima, or points of inflection.

Sequences

Sequences questions ask you to spot a pattern and turn it into a rule.

1. Generate terms from a term-to-term rule (each term built from the previous one) or a position-to-term rule (each term built directly from its position number).
2. Recognise triangular, square and cube number sequences, and simple arithmetic progressions.
3. Find the nth term of a linear sequence, and at Extension, a quadratic sequence.

Common mistakes in Algebra

How to revise this properly

Do not just read this page once and move on. Build your own oxfordaqa igcse mathematics revision notes by rewriting each worked example above from a blank page, checking your version against this one only after you have committed to an answer. That is what separates oxfordaqa igcse mathematics notes that actually work from a highlighted printout nobody opens again. Once you can do every example here without hesitation, move to oxfordaqa igcse mathematics practice questions from full past papers, where algebra rarely appears in isolation and is instead folded into geometry, ratio or graph-reading questions.

Self-check questions

1. Expand and simplify (2x - 1)(x + 4).
2. Factorise x² - 9x + 20.
3. Solve 4(x - 2) = 3x + 5.
4. Solve the simultaneous equations 2x + y = 11 and x - y = 1.
5. Solve the inequality -3x + 6 > 0, and state whether the boundary is included.
6. Find the nth term of the sequence 7, 12, 17, 22, ...
7. Find the gradient of the line joining (1, 2) and (4, 11).
8. A straight line has gradient 4 and passes through (2, 5). Find its equation in the form y = mx + c.
9. Differentiate y = 4x³ - 2x, and find the value of dy/dx when x = 1.

Every one of these is algebra oxfordaqa igcse content you should be able to do without a calculator, in under two minutes each. If any took longer, that is your revision priority this week, not next month. Treat speed as part of the skill, not a separate concern from accuracy, because a timed paper punishes slow-but-correct just as hard as it punishes fast-but-wrong.