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Edexcel IGCSE Maths (4MA1) · Revision notes · Algebraic Modelling

IGCSE Maths Algebra Revision Notes

Short, plain-English notes on solving equations & fractions for Pearson Edexcel International GCSE Mathematics A (4MA1), Higher tier. Each of the 11 sub-topics has what you need to know and one example with its answer. Cover the answer and try the example first, then practise the topic with worked solutions.

Algebraic Expressions

Simplifying, collecting terms, expanding brackets, factoring common terms.

Simplify by collecting like terms; expand brackets before combining; factor common terms out.

Example: Simplify 4x + 5y − 2(x − 3y).

Answer: = 4x + 5y − 2x + 6y = 2x + 11y.

Practise it: Algebraic Expressions questions and answers

STEM Formulae

Substituting negative values / fractions and rearranging basic variables.

A ‘STEM formula’ is a science / engineering formula written in symbols. Substitute known values then evaluate.

Example: Given v² = u² + 2as, find v when u=5, a=3, s=6 (v>0).

Answer: v² = 25 + 36 = 61 → v ≈ 7.81.

Practise it: Rearranging Formulae questions and answers

Linear Equations & Inequalities

Solving multi-step equations, plotting inequalities on number lines.

An inequality replaces ‘=’ with ‘<, >, ≤, ≥’. Solve like an equation — but flip the inequality if you multiply/divide by a negative.

Example: Solve 4x − 5 ≥ 11.

Answer: 4x ≥ 16 → x ≥ 4.

Practise it: Inequalities questions and answers · Linear Equations questions and answers

Expanding Brackets

Expanding and simplifying double-bracket expressions (quadratic expansion).

‘Double brackets’ (e.g. (x+a)(x+b)) expand to a quadratic using FOIL.

Example: Expand and simplify (x + 4)(x + 7).

Answer: = x² + 7x + 4x + 28 = x² + 11x + 28.

Practise it: Expanding and Factorising questions and answers

Factorising Quadratics

Factoring trinomials of the form x² + bx + c.

Factorising x² + bx + c means finding two brackets that multiply back to the quadratic.

Example: Factorise x² − 8x + 15.

Answer: Find two numbers ×15 and +(−8): −3 and −5. So (x − 3)(x − 5).

Practise it: Expanding and Factorising questions and answers · Quadratic Equations questions and answers

Linear Simultaneous Equations

Solving two-variable linear systems algebraically and graphically.

Simultaneous equations are two equations in two unknowns. Solve by elimination, substitution, or graphically.

Example: Solve x + y = 7 and 2x − y = 5.

Answer: Add: 3x = 12 → x = 4, y = 3.

Practise it: Simultaneous Equations questions and answers

Quadratic Equations

Solving ax² + bx + c = 0 by factorisation, completing the square or the quadratic formula.

A quadratic equation is ax² + bx + c = 0. Solve by factorising, completing the square, or the quadratic formula.

Example: Solve x² + 2x − 24 = 0 by factorisation.

Answer: (x + 6)(x − 4) = 0 → x = −6 or x = 4.

Practise it: Quadratic Equations questions and answers

Completing the Square

Completing the square for simple expressions of the form x² + bx.

Completing the square rewrites x² + bx as (x + b/2)² − (b/2)² — useful for finding the vertex of a parabola.

Example: Complete the square for x² + 6x.

Answer: = (x + 3)² − 9.

Practise it: Completing the Square questions and answers

Non-linear Simultaneous Equations

Solving one linear and one quadratic equation simultaneously by substitution.

With one linear and one quadratic equation, rearrange the linear one and substitute it into the quadratic — you get a single quadratic to solve.

Example: Solve y = x + 1 and x² + y² = 25.

Answer: x² + (x + 1)² = 25 → x² + x − 12 = 0 → x = 3 or −4 → (3, 4) and (−4, −3).

Practise it: Simultaneous Equations questions and answers

Quadratic Inequalities

Solving quadratic inequalities by finding the roots and sketching the parabola.

For a quadratic inequality, find the roots, sketch the parabola, then read off where the curve is above (> 0) or below (< 0) the x-axis.

Example: Solve x² − x − 6 ≤ 0.

Answer: (x − 3)(x + 2) ≤ 0 → −2 ≤ x ≤ 3.

Practise it: Inequalities questions and answers

Algebraic Proof

Proving results with algebra (e.g. consecutive / odd integers) and disproving by counterexample.

An algebraic proof uses letters for general numbers, e.g. 2n for any even number and 2n + 1 for any odd number, then simplifies to show the result always holds.

Example: Prove the sum of two consecutive odd numbers is a multiple of 4.

Answer: (2n + 1) + (2n + 3) = 4n + 4 = 4(n + 1).

Practise it: Proof questions and answers

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