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GCSE Maths Rearranging Formulae Notes

Rearranging a formula means making a different letter the subject. Use inverse operations in reverse order, just like solving an equation.

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Key concepts

Subject of a formula
The letter on its own, usually before the equals sign, e.g. y is the subject of y = 3x + 2.
Inverse operations
Undo operations in reverse order: undo + and − first, then × and ÷, then powers and roots.
Subject on both sides
Collect the required letter on one side, factorise it out, then divide.

Key formulas

Undo a square

if x² = k then x = ±√k

Undo a square root

if √x = k then x = k²

Worked example

Make r the subject of A = πr².

  1. 1Divide both sides by π: A/π = r².
  2. 2Square root both sides: √(A/π) = r.
  3. 3Write r as the subject: r = √(A/π).

Answer: r = √(A/π)

Common mistakes

✗ Square rooting only one term, e.g. √(a² + b²) = a + b.

✓ You cannot split a root over addition — a² + b² must be combined first.

✗ Forgetting to apply the operation to every term.

✓ Whatever you do, do it to the whole of both sides.

✗ Not factorising when the subject appears twice.

✓ Collect the terms containing the letter, factorise it out, then divide.

Quick check

Try these, then reveal the answers.

Make x the subject of y = 2x + 5.Reveal answer

x = (y − 5)/2

Make t the subject of v = u + at.Reveal answer

t = (v − u)/a

Make x the subject of y = x².Reveal answer

x = ±√y

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Exam tips

  • Think of it as solving for a letter — the same inverse operations apply.
  • When the new subject appears twice, factorising is the key step examiners reward.
  • Leave answers as neat fractions or surds rather than rounding.

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Key terms explained

Short, exam-ready definitions with a formula, a worked example and an examiner tip.

What is area under a curve?What is chain rule?What is completing the square?What is definite integral?What is differentiation?What is discriminant?Full glossary

Frequently asked questions

How is rearranging a formula different from solving an equation?+

The method is identical — inverse operations — but the answer is a formula in the other letters, not a single number.

What do I do when the subject is inside a square root?+

Square both sides to remove the root, then continue rearranging as normal.

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