Rearranging Complex Formulae –

Rearranging complex formulae is a Higher tier GCSE Maths skill that goes beyond basic rearrangement. When the subject appears more than once, or the formula involves powers and roots, you need additional techniques including factorising.

What Is Rearranging Complex Formulae?

A complex rearrangement is one where the target variable appears more than once in the formula, or where it is trapped inside a square, square root, or fraction. Simple inverse operations are not enough on their own — you also need to collect terms and factorise.

For example, to make x the subject of y = (ax + b) / (cx + d), you must multiply out, collect all terms containing x on one side, factorise x out, and then divide. This factorisation step is the key difference from basic rearrangement.

Another common type involves making a variable the subject when it is squared (such as making r the subject of A = πr²) or under a square root. You apply the appropriate inverse — square-rooting or squaring — at the right stage of the process.

Key Formulas

Step-by-Step Method

1. If there is a fraction, multiply both sides by the denominator to clear it.
2. Expand any brackets so all terms are visible.
3. Collect all terms containing the target variable on one side and all other terms on the other.
4. Factorise the target variable out of the collected terms.
5. Divide both sides by the remaining bracket to isolate the target variable.

Worked Example 1 — Foundation Level

Question: This is a Higher only topic. Here is an accessible entry. Make r the subject of A = πr².

Working:

Step 1 — Divide both sides by π: A / π = r².

Step 2 — Take the positive square root (r is a length): r = √(A / π).

Answer: r = √(A / π)

Worked Example 2 — Higher Level

Question: Make x the subject of y = (5x + 3) / (2x - 1).

Working:

Step 1 — Multiply both sides by (2x - 1): y(2x - 1) = 5x + 3.

Step 2 — Expand: 2xy - y = 5x + 3.

Step 3 — Collect x terms on one side: 2xy - 5x = 3 + y.

Step 4 — Factorise x: x(2y - 5) = 3 + y.

Step 5 — Divide: x = (3 + y) / (2y - 5).

Answer: x = (3 + y) / (2y - 5)

Worked Example 3 — Exam Style

Question: Make t the subject of v = u + at. Hence make a the subject of s = ut + ½at². (5 marks)

Working:

Part 1 — v = u + at: subtract u, then divide by a: t = (v - u) / a.

Part 2 — s = ut + ½at²: this is harder because a appears only once but t appears twice.

Step 1 — Factor out t from the right is not helpful here, so rearrange for a instead.

Step 2 — Subtract ut: s - ut = ½at².

Step 3 — Multiply by 2: 2(s - ut) = at².

Step 4 — Divide by t²: a = 2(s - ut) / t².

Answer: t = (v - u) / a; a = 2(s - ut) / t²

Common Mistakes

• Forgetting to factorise when the subject appears twice. If you have 2xy - 5x on one side, you must write x(2y - 5) before dividing. Dividing without factorising gives an incorrect result.
• Square-rooting only one side. When taking a square root, it must be applied to both sides of the equation, and the right-hand side often needs to remain under a single root.
• Losing the ± symbol. In pure algebra, x² = k gives x = ±√k. In context, you may only need the positive root, but in general both solutions exist.

Exam Tips

• If you see the target variable in two places, your plan should be: collect, factorise, divide.
• Write out each step on a separate line — method marks are awarded at each stage.
• After rearranging, substitute numbers to check: pick simple values, work out both sides, and see if they match.

Practice Questions

Q1 (Higher): Make h the subject of V = πr²h.

Q2 (Higher): Make x the subject of a = (x + b) / (x - c).

Q3 (Higher): Make p the subject of q = √(3p + 1).

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Related Topics

• Rearranging Formulae
• Factorising Expressions
• Solving Linear Equations

Summary

• When the target variable appears twice, collect those terms, factorise the variable out, then divide.
• For squares, take the square root; for square roots, square both sides.
• Clear fractions early by multiplying by the denominator.
• Always expand brackets so you can see every occurrence of the target variable.
• Check your rearrangement by substituting simple values into both the original and new formula.