Algebraic Manipulation

This guide covers subtopics 2.1 to 2.3 of Topic 2, Algebra and graphs, for Cambridge O Level Mathematics (Syllabus D) 4024, 2025–2027 series.

Where this fits in 4024

Algebraic manipulation is the working vocabulary for almost everything that follows in 4024 — equations (2.5), inequalities (2.6), graphs of functions (2.10), and later, coordinate geometry and mensuration problems that reduce to an algebraic equation once the geometry is stripped away. Before any of that is worth attempting, you need to simplify, expand, factorise and rearrange algebraic expressions confidently and without a calculator.

A note on scope. This page is written specifically against Cambridge O Level Mathematics 4024, which has one flat, untiered set of outcomes. Cambridge IGCSE Mathematics 0580 covers algebra under the same topic name but splits it into Core and Extended tiers with different subtopic numbering, and its Core tier does not include everything below (completing the square and the full set of factorising forms are Extended-only at IGCSE). If you’re studying 0580, check your tier before assuming this page matches your syllabus exactly — a dedicated IGCSE resource is planned separately.

Syllabus coverage

CAMBRIDGE O LEVEL MATHEMATICS (SYLLABUS D) 4024

• Know that letters can represent generalised numbers, and substitute numbers into expressions and formulas (2.1)
• Simplify expressions by collecting like terms (2.2)
• Expand products of algebraic expressions, including products of more than two brackets (2.2)
• Factorise by extracting common factors (2.2)
• Factorise expressions of the form ax + bx + kay + kby, a²x² − b²y², a² + 2ab + b², ax² + bx + c, and ax³ + bx² + cx (2.2)
• Complete the square for expressions of the form ax² + bx + c (2.2)
• Manipulate algebraic fractions, including factorising and simplifying rational expressions (2.3)

4024 is not tiered — every candidate is assessed on all of the above, across Paper 1 and Paper 2.

Introduction to algebra

Algebra uses letters to stand for numbers whose value isn’t fixed, or isn’t known yet. Two things you’ll do constantly:

• Substitute — replace a letter with a given number and work out the result. If V = IR and I = 4, R = 6, then V = 4 × 6 = 24.
• Generalise — write an expression that works for any number, not just one example. “The product of two consecutive even numbers” becomes n(n + 2), where n is any even number.

Simplifying by collecting like terms

Like terms have exactly the same letters raised to exactly the same powers — 3a² and 5a² are like terms; 3a² and 3a are not. To simplify, add or subtract the coefficients of like terms and leave everything else unchanged.

Worked example. Simplify 2a² + 3ab − 1 + 5a² − 9ab + 4.

a² terms:  2a² + 5a²  = 7a²
ab terms:  3ab − 9ab  = −6ab
numbers:   −1 + 4      = 3

answer: 7a² − 6ab + 3

Expanding products

Expanding means multiplying out brackets. Multiply every term inside the bracket by whatever sits outside it — and when two brackets are multiplied together, every term in the first multiplies every term in the second.

Worked example. Expand 3x(2x − 4y).

3x(2x − 4y) = 3x × 2x − 3x × 4y = 6x² − 12xy

Worked example. Expand (3x + y)(x − 4y).

(3x + y)(x − 4y)
= 3x·x + 3x·(−4y) + y·x + y·(−4y)
= 3x² − 12xy + xy − 4y²
= 3x² − 11xy − 4y²

4024 also expects products of more than two brackets, expanded one pair at a time.

Worked example. Expand (x − 2)(x + 3)(2x + 1).

Step 1 — expand the first two brackets:
(x − 2)(x + 3) = x² + 3x − 2x − 6 = x² + x − 6

Step 2 — multiply that result by the third bracket:
(x² + x − 6)(2x + 1)
= 2x³ + x² + 2x² + x − 12x − 6
= 2x³ + 3x² − 11x − 6

Factorising

Factorise means the reverse of expanding — write an expression as a product of brackets, taken as far as it will go (“factorise fully”).

Common factors. Look for the highest common factor of every term and take it outside a bracket.

9x² + 15xy = 3x(3x + 5y)

Grouping — ax + bx + kay + kby. Group terms in pairs that share a common factor, factorise each pair, then take out the bracket they now share.

ax + bx + kay + kby
= x(a + b) + ky(a + b)
= (a + b)(x + ky)

Difference of two squares — a²x² − b²y². This always factorises as the product of a sum and a difference.

a²x² − b²y² = (ax − by)(ax + by)

Perfect square — a² + 2ab + b². This is a squared bracket in disguise.

a² + 2ab + b² = (a + b)²

Quadratic trinomial — ax² + bx + c. Find two numbers that multiply to give ac and add to give b, then split the middle term and group.

x² + 5x + 6:  need two numbers multiplying to 6, adding to 5 → 2 and 3
x² + 5x + 6 = (x + 2)(x + 3)

Common factor plus quadratic — ax³ + bx² + cx. Take out the common factor first, then factorise what’s left if it will factorise further.

2x³ + 10x² + 12x
= 2x(x² + 5x + 6)
= 2x(x + 2)(x + 3)

Completing the square

Writing ax² + bx + c as a squared bracket plus (or minus) a constant makes it possible to solve equations, find a minimum or maximum, and sketch a graph without plotting points. For x² + bx + c (coefficient of x² equal to 1), halve the coefficient of x, and correct for the difference between the square you’ve made and the original expression. When the coefficient of x² is not 1, factor it out of the x² and x terms first, complete the square inside the bracket, then multiply back out.

Worked example. Write x² + 6x + 5 in completed square form.

half of 6 is 3, so start from (x + 3)²
(x + 3)² = x² + 6x + 9

that has 9 where the original only has 5, so subtract 4:
x² + 6x + 5 = (x + 3)² − 4

Check by expanding: (x + 3)² − 4 = x² + 6x + 9 − 4 = x² + 6x + 5. ✓

Worked example (leading coefficient not 1). Write 2x² + 8x + 3 in completed square form.

factor 2 out of the x² and x terms only:
2x² + 8x + 3 = 2(x² + 4x) + 3

complete the square inside the bracket:
half of 4 is 2, so (x + 2)² = x² + 4x + 4
x² + 4x = (x + 2)² − 4

substitute back and multiply the −4 by the 2 that was factored out:
2x² + 8x + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 8 + 3 = 2(x + 2)² − 5

Check by expanding: 2(x + 2)² − 5 = 2(x² + 4x + 4) − 5 = 2x² + 8x + 8 − 5 = 2x² + 8x + 3. ✓

Algebraic fractions

The rules are the same as for ordinary fractions — but factorising first is usually what makes a fraction simplify at all.

Adding fractions with different denominators. Find a common denominator, then combine the numerators.

x/3 + (x − 4)/2
= 2x/6 + 3(x − 4)/6
= (2x + 3x − 12)/6
= (5x − 12)/6

Multiplying and dividing. Multiply numerators together and denominators together; to divide, multiply by the reciprocal of the second fraction.

3a/4 × 9a/10 = 27a²/40

3a/4 ÷ 9a/10 = 3a/4 × 10/9a = 30a/36a = 5/6

Simplifying rational expressions. Factorise the numerator and denominator first, then cancel any factor common to both.

Worked example. Simplify (x² − 2x) / (x² − 5x + 6).

numerator:   x² − 2x = x(x − 2)
denominator: x² − 5x + 6 = (x − 2)(x − 3)

(x² − 2x) / (x² − 5x + 6) = x(x − 2) / (x − 2)(x − 3) = x / (x − 3),  x ≠ 2, 3

Cancelling is only ever valid for a factor of the whole numerator and the whole denominator — never for a term that’s merely added or subtracted somewhere inside an unfactorised expression.

Common mistakes

• Cancelling terms instead of factors. (x + 4)/x is not equal to 4 — you can only cancel something that multiplies the entire numerator and the entire denominator, which means factorising first.
• Forgetting to multiply every term when expanding a bracket, especially the sign of the last term in a product like (x − 2)(x + 3).
• Losing a negative sign when subtracting one bracket from another, e.g. treating 2x − 3(x − 5) as 2x − 3x − 15 instead of 2x − 3x + 15.
• Stopping a factorisation too early. “Factorise” in 4024 always means factorise fully — 2x² + 4x should become 2x(x + 2), not just x(2x + 4).
• Getting the sign wrong when completing the square, e.g. writing x² + 6x + 5 = (x + 3)² + 4 instead of (x + 3)² − 4. Expanding your answer back out is the fastest way to check.

Quick revision checklist

• Collecting like terms to simplify an expression
• Expanding a single bracket, a product of two brackets, and a product of three brackets
• Factorising by common factor, by grouping, as a difference of two squares, as a perfect square, and as a quadratic trinomial
• Completing the square for x² + bx + c
• Adding, multiplying, dividing and simplifying algebraic fractions, including factorising before cancelling

Related resources

Once expanding, factorising and completing the square feel automatic, they carry straight into 2.5 Equations — solving quadratic equations by factorisation, by completing the square, and with the quadratic formula all build directly on the skills above.

• Indices and Equations — subtopics 2.4-2.5, continuing directly from this page
• Cambridge O Level Mathematics subject hub

Written against Cambridge O Level Mathematics (Syllabus D) 4024, 2025–2027 series. Always check the current syllabus for your examination year.