What should I learn from this Further Maths lesson?

By the end of this lesson, you should be able to Substitute numerical values into formulae and expressions to calculate results; Rearrange formulae to change the subject, including cases requiring multiple steps; Expand single and double brackets and simplify the resulting expressions; Factorise expressions including common factors and quadratic expressions.

How should I revise Formulae and Expressions: Use and Manipulation?

Read each section, then close the notes and explain the idea from memory. Check your explanation against the page and finish with an exam question from Algebra.

Formulae and Expressions: Use and Manipulation

Formulae and Expressions: Use and Manipulation is part of Use and manipulation of formulae and expressions in GCSE AQA Higher Further Maths.

What this lesson covers

Formulae and Expressions: Use and Manipulation: Learn how to use and manipulate formulae and expressions, including rearranging and simplifying them.

Substitute numerical values into formulae and expressions to calculate results
Rearrange formulae to change the subject, including cases requiring multiple steps
Expand single and double brackets and simplify the resulting expressions
Factorise expressions including common factors and quadratic expressions

Key ideas to keep in view

Use this lesson to stay inside Use and manipulation of formulae and expressions while connecting the detail back to the wider Algebra topic.

Algebra
Use and manipulation of formulae and expressions
GCSE AQA Higher Further Maths

Revision notes

Read the core explanations from Formulae and Expressions: Use and Manipulation before testing yourself from memory.

1. Why Formulae and Expressions Matter

Every time you calculate speed, area, or interest, you are using a formula. The ability to substitute into, rearrange, expand, and factorise algebraic expressions is one of the most frequently tested skills across all GCSE Maths papers.

Think of a formula as a machine: you feed in known values, and it produces the answer you need. Rearranging the formula is like rewiring the machine so a different quantity comes out. Both skills appear on Paper 1 (non-calculator) and Papers 2–3 (calculator), so you need to be confident with the algebra regardless of whether a calculator is available.

2. Substituting into Formulae and Expressions

Substitution means replacing each letter in a formula with its given numerical value, then evaluating the result. The key discipline is to write the substitution out explicitly before you calculate — this earns method marks even if arithmetic slips occur later.

A common source of errors is negative values. When substituting a negative number, always wrap it in brackets: write $(-3)^2$, not $-3^2$. Without brackets, $-3^2 = -9$, but $(-3)^2 = 9$ — two very different results.

Worked example: Substituting into a Formula

The formula for the area of a trapezium is $A = \frac{1}{2}(a + b)h$. Find $A$ when $a = 5$, $b = 9$, and $h = 4$.

$A = \frac{1}{2}(a + b)h$ — Always start by stating the formula — this earns a method mark [M1].
$A = \frac{1}{2}(5 + 9) \times 4$ — Replace $a$, $b$, and $h$ with their given values [M1].
$5 + 9 = 14$, so $A = \frac{1}{2} \times 14 \times 4$ — Evaluate the bracket first, following order of operations.
$A = \frac{1}{2} \times 56 = 28$ — $14 \times 4 = 56$, then $56 \div 2 = 28$. Final answer with units [A1].

Answer: $A = 28 \text{ (square units)}$

3. Rearranging Formulae

Rearranging a formula means isolating a different variable — changing the subject. The method mirrors solving an equation: whatever you do to one side, you do to the other. Work outward from the variable you want, undoing operations in reverse order.

Worked example: Rearranging a Formula — One Step

Make $u$ the subject of $v = u + at$.

$v = u + at$ — Identify which operation connects $u$ to the rest: $at$ is added to $u$.
$v - at = u$ — Subtracting $at$ from both sides isolates $u$ [M1].
$u = v - at$ — Convention is to write the subject on the left. This is the rearranged formula [A1].

Answer: $u = v - at$

Worked example: Rearranging a Formula — Multiple Steps

Make $r$ the subject of $A = \pi r^2$.

$A = \pi r^2$ — Two operations act on $r$: it is squared, then multiplied by $\pi$. Undo them in reverse order.
$\frac{A}{\pi} = r^2$ — Dividing by $\pi$ undoes the multiplication [M1].
$r = \sqrt{\frac{A}{\pi}}$ — Taking the square root undoes the squaring. Since $r$ is a length, we take the positive root [A1].

Answer: $r = \sqrt{\frac{A}{\pi}}$

4. Expanding Brackets and Simplifying Expressions

Expanding brackets removes the bracket by multiplying every term inside by the term outside. For a single bracket, multiply each term inside by the factor outside. For double brackets, every term in the first bracket multiplies every term in the second.

Worked example: Expanding Double Brackets

Expand and simplify $(2x + 3)(x - 5)$.

$2x \times x = 2x^2$ — First terms from each bracket.
$2x \times (-5) = -10x$ — Outer terms — carry the negative sign carefully.
$3 \times x = 3x$ — Inner terms.
$3 \times (-5) = -15$ — Last terms — again, carry the negative sign.
$2x^2 - 10x + 3x - 15 = 2x^2 - 7x - 15$ — $-10x + 3x = -7x$. Final simplified expression [A1].

Answer: $2x^2 - 7x - 15$

5. Factorising Expressions

Factorising is the reverse of expanding — you rewrite an expression as a product of factors. There are three main types you need at GCSE: taking out a common factor, factorising quadratics of the form $x^2 + bx + c$, and factorising quadratics of the form $ax^2 + bx + c$.

Always check for a common factor first. If every term shares a factor, take it outside a bracket before attempting any other method. This keeps the numbers smaller and reduces errors in later steps.

Worked example: Factorising — Common Factor

Factorise $6x^2 - 9x$ fully.

$\text{HCF of } 6x^2 \text{ and } 9x = 3x$ — Both terms share the factor $3x$.
$6x^2 \div 3x = 2x$ and $9x \div 3x = 3$ — These become the terms inside the bracket.
$6x^2 - 9x = 3x(2x - 3)$ — Check by expanding: $3x \times 2x = 6x^2$ and $3x \times (-3) = -9x$ ✓ [A1].

Answer: $3x(2x - 3)$

Worked example: Factorising a Quadratic — $x^2 + bx + c$

Factorise $x^2 + 7x + 12$.

Need two numbers: product $= 12$, sum $= 7$ — List factor pairs of 12: $(1, 12)$, $(2, 6)$, $(3, 4)$. Check sums: $3 + 4 = 7$ ✓
$x^2 + 7x + 12 = (x + 3)(x + 4)$ — Place the two numbers into brackets [A1].
$(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$ ✓ — Always expand to check — this takes 10 seconds and prevents dropped marks.

Answer: $(x + 3)(x + 4)$

Worked example: Factorising a Quadratic — $ax^2 + bx + c$

Factorise $2x^2 + 7x + 3$.

$a \times c = 2 \times 3 = 6$ — Find two numbers with product $6$ and sum $7$ (the value of $b$).
$1 \times 6 = 6$ and $1 + 6 = 7$ ✓ — The pair is $1$ and $6$.
$2x^2 + x + 6x + 3$ — Rewrite $7x$ as $x + 6x$ using the pair found.
$x(2x + 1) + 3(2x + 1)$ — Factor from each pair: $x$ from the first two terms, $3$ from the last two.
$(2x + 1)(x + 3)$ — $(2x + 1)$ is the common bracket [A1]. Verify: $(2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3$ ✓

Answer: $(2x + 1)(x + 3)$

6. Putting It All Together — Exam Practice

Exam questions on formulae and expressions often chain several skills together. You might need to substitute, then rearrange, or expand and then factorise. Recognising which skill is needed — and in which order — is what separates a confident answer from a confused one.

7. Exam Technique and Common Mistakes

The most reliable way to lose marks on these topics is through sign errors and skipped steps. Examiners award method marks for correct working even when the final answer is wrong — but only if the working is visible. Write every substitution and every algebraic step on a separate line.

Sources and content standards

Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-14.

AQA Level 2 Certificate in Further Mathematics (8365) specification How we check content Report a correction — include this lesson URL

Formulae and Expressions: Use and Manipulation Revision Notes | GCSE AQA Higher Further Maths