Algebraic manipulation

Edexcel International GCSE Mathematics A· 2.2 (2016 spec)· 25 min read

1. Foundation: Core Algebraic Manipulation Skills★★☆☆☆⏱ 7 min

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📘 Definition

Like terms

Algebraic terms that have identical variable parts (same letters raised to the same powers)

All Foundation tier manipulation starts with four core skills: substitution, collecting like terms, expanding single brackets, and factorising common factors. These form the base for all more complex algebraic work.

📐 Worked Example

Evaluate $3 x^{2} - 2 y + 7$ when $x = 2$ and $y = - 4$

1
Substitute the given values into the expression
$3(2)^2 - 2(-4) + 7$
2
Calculate exponents first per BODMAS rules
$3(4) + 8 + 7$
3
Simplify to get the final result
$12 + 8 + 7 = 27$

📐 Worked Example

Simplify $4 x + 3 y - 2 x + 7 y - 9$

1
Group like terms together
$(4x - 2x) + (3y + 7y) - 9$
2
Combine like terms to get the simplified expression
$2x + 10y - 9$

📐 Worked Example

Expand $3 x (2 x + 5)$

1
Multiply the term outside the bracket by each term inside the bracket
$3x \times 2x + 3x \times 5$
2
Simplify each product
$6x^2 + 15x$

📐 Worked Example

Fully factorise $8 x y + 12 y^{2}$

1
Find the highest common factor (HCF): HCF of 8 and 12 is 4, HCF of $x y$ and $y^{2}$ is $y$ , so overall HCF is $4 y$
2
Factor out the HCF from both terms
$4y(2x + 3y)$

Exam tip:

Always check you have taken out the highest common factor, not just any common factor, to get full marks for factorisation questions.

2. Foundation: Quadratic Expansion and Factorisation★★★☆☆Foundation / Higher only⏱ 6 min

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📘 Definition

Monic quadratic

A quadratic expression where the coefficient of the $x^{2}$ term is equal to 1

Foundation tier requires you to expand products of two linear expressions and factorise monic quadratic expressions. The FOIL method is a reliable way to expand two brackets without missing terms.

📐 Worked Example

Expand and simplify $(x + 8) (x - 5)$

1
Apply the FOIL method
$x \times x + x \times (-5) + 8 \times x + 8 \times (-5)$
2
Simplify each product
$x^2 -5x +8x -40$
3
Collect like terms
$x^2 + 3x -40$

📐 Worked Example

Factorise $x^{2} + 10 x + 24$

1
Find two numbers that multiply to 24 and add to 10: 6 and 4
2
Write the factorised form using these two numbers
$(x + 6)(x + 4)$

Exam tip:

After factorising a quadratic, expand it back immediately to check you get the original expression, to catch sign errors early.

3. Higher: Advanced Expansion and Factorisation★★★★☆Higher only⏱ 5 min

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Higher tier adds expansion of products of 3 or more linear expressions, and factorisation of non-monic quadratic expressions (where the coefficient of $x^{2}$ is not 1).

📐 Worked Example

Expand and simplify $(x + 2) (x + 3) (x - 1)$

1
First expand the first two brackets
$(x + 2)(x + 3) = x^2 + 5x + 6$
2
Multiply the result by the third bracket
$(x^2 +5x +6)(x -1) = x^3 -x^2 +5x^2 -5x +6x -6$
3
Collect like terms
$x^3 +4x^2 +x -6$

📐 Worked Example

Fully factorise $6 x^{2} - 5 x - 6$

1
Multiply the $x^{2}$ coefficient and the constant term: $6 \times (- 6) = - 36$
2
Find two numbers that multiply to -36 and add to -5: -9 and 4
3
Split the middle term using these numbers
$6x^2 -9x +4x -6$
4
Factorise in pairs, then extract the common bracket
$3x(2x -3) + 2(2x -3) = (3x + 2)(2x - 3)$

Exam tip:

When expanding three or more brackets, always expand two first, then multiply by the remaining bracket to avoid missing terms.

4. Higher: Algebraic Fractions and Completing the Square★★★★☆Higher only⏱ 5 min

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Two key Higher tier skills are simplifying algebraic fractions and rewriting quadratics in completed square form. Always factorise numerators and denominators first when manipulating algebraic fractions.

📐 Worked Example

Simplify $\frac{x ^{2} + 7 x + 12}{x ^{2} - 9}$

1
Factorise the numerator
$x^2 +7x +12 = (x + 3)(x + 4)$
2
Factorise the denominator (difference of two squares)
$x^2 -9 = (x + 3)(x - 3)$
3
Cancel the common factor of $(x + 3)$ (where $x \neq = - 3$ )
$\frac{x + 4}{x - 3}$

📐 Worked Example

Write $2 x^{2} + 6 x - 1$ in the form $a (x + b)^{2} + c$

1
Factor out the $x^{2}$ coefficient from the first two terms
$2(x^2 + 3x) - 1$
2
Complete the square inside the bracket
$x^2 + 3x = (x + \frac{3}{2})^2 - \frac{9}{4}$
3
Substitute back and simplify
$2\left[(x + \frac{3}{2})^2 - \frac{9}{4}\right] -1 = 2(x + \frac{3}{2})^2 - \frac{9}{2} - 1 = 2(x + \frac{3}{2})^2 - \frac{11}{2}$

Exam tip:

For completing the square questions, always expand your final answer back to the original quadratic form to catch arithmetic errors.

5. Higher: Simple Algebraic Proof★★★★★Higher only⏱ 3 min

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Higher tier requires you to use algebra to construct simple proofs, for example to show that a sequence is always even, or that the product of two odd numbers is odd. Always start proofs by clearly defining your variables.

📐 Worked Example

Prove that the product of two consecutive even numbers is always divisible by 8

1
Define variables: let $n$ be an integer, so the first even number is $2 n$ , the next consecutive even number is $2 n + 2$
2
Calculate the product
$2n(2n + 2) = 4n(n + 1)$
3
$n$ and $n + 1$ are consecutive integers, so one of them is even: $n (n + 1) = 2 k$ where $k$ is an integer
4
Substitute back: product $= 4 (2 k) = 8 k$ , which is divisible by 8, as required

Exam tip:

End all proofs with a clear concluding statement confirming you have proven the required result, to make your argument complete.

6. Common Pitfalls

Wrong move:

Forgetting negative signs when expanding brackets, e.g. writing $2 x (x - 3) = 2 x^{2} - 6$ instead of $2 x^{2} - 6 x$

Why:

Rushing multiplication and omitting variables or sign changes

Correct move:

Multiply the term outside the bracket by every single term inside, including signs, then check each term after expansion.

Wrong move:

Partially factorising expressions, e.g. writing $12 x^{2} + 18 x = 2 x (6 x + 9)$ instead of $6 x (2 x + 3)$

Why:

Not identifying the highest common factor of both coefficients and variables

Correct move:

Calculate the HCF of numerical coefficients separately from the HCF of variable terms, then combine them for the full common factor.

Wrong move:

Sign errors when factorising quadratics with negative constants, e.g. writing $x^{2} - x - 6 = (x - 2) (x + 3)$ instead of $(x - 3) (x + 2)$

Why:

Mixing up the signs of the two factors when their sum is negative

Correct move:

After factorising, expand the result immediately to verify it matches the original expression.

Wrong move:

Cancelling terms instead of factors in algebraic fractions, e.g. simplifying $\frac{x + 2}{x + 4}$ as $\frac{2}{4} = \frac{1}{2}$

Why:

Confusing terms (parts added together) with factors (parts multiplied together)

Correct move:

Always factorise numerator and denominator fully first, then cancel only common factors, not individual terms.

Wrong move:

Forgetting to multiply the subtracted constant term by $a$ when completing the square for non-monic quadratics, e.g. writing $2 x^{2} + 4 x = 2 (x + 1)^{2} - 1$ instead of $2 (x + 1)^{2} - 2$

Why:

Not accounting for the factor $a$ outside the bracket when expanding the completed square

Correct move:

After completing the square inside the bracket, distribute the $a$ factor to all terms inside before simplifying constants.

7. Quick Reference Cheatsheet

Skill	Foundation Requirement	Higher Requirement	Key Exam Tip
Substitution	Evaluate with positive/negative integers	Same as Foundation	Follow BODMAS order when calculating
Collecting like terms	Simplify linear expressions with up to 2 variables	Same as Foundation	Group terms by identical variable parts first
Expand brackets	Single brackets, 2 linear brackets	3+ linear brackets	Use FOIL for 2 brackets, expand 2 first for 3+
Factorisation	Common factors, monic quadratics	Common factors, monic + non-monic quadratics	Always take out the highest common factor first
Algebraic fractions	Not required	Simplify, add/subtract linear/quadratic fractions	Factorise numerator and denominator before cancelling
Completing the square	Not required	Rewrite $a x^{2} + b x + c$ as $a (x + b)^{2} + c$	Expand your result to check for errors
Algebraic proof	Not required	Construct simple integer proofs	Start by defining variables clearly e.g. let $n$ be integer

8. Frequently Asked

Do I need to factorise expressions fully in the exam?

Yes, always take out all common numerical and variable factors before checking for further factorisation (e.g. quadratic factors). Partially factorised answers will not receive full marks.

Can I use the FOIL method to expand two linear brackets?

Yes, FOIL (First, Outer, Inner, Last) is a valid method for expanding products of two linear expressions, as long as you collect like terms correctly after expansion.

What is the difference between Foundation and Higher tier requirements for this topic?

Foundation tier only requires manipulation of monic quadratics, expansion of up to two linear brackets, and no algebraic fractions, completing the square or proof. Higher tier adds all these additional skills.

Going deeper

official_specEdexcel IGCSE Maths A (4MA1) SpecificationSee section 2.2 for full algebraic manipulation requirements

What's Next

Now that you have mastered algebraic manipulation, you are ready to apply these skills to solving equations, the next core topic in the Edexcel IGCSE Maths A syllabus. All equation-solving skills, from linear equations to quadratic equations and simultaneous equations, rely on the manipulation skills you have learned here. For Foundation tier students, focus on practicing monic quadratic factorisation and expanding two brackets to prepare for solving quadratic equations in section 2.7. For Higher tier students, make sure you are confident with non-monic factorisation, completing the square and algebraic fractions, as these are essential for solving harder quadratics, rearranging complex formulae, and tackling algebraic proof questions in later topics. Regular practice of mixed manipulation questions will help you avoid common sign and factorisation errors in your exam.