Rearranging Harder Formulae — GCSE Maths

At GCSE, rearranging formulae goes beyond simple one- or two-step moves. You may need to deal with a formula where the subject appears twice, where it is trapped inside a square root, or where squaring is needed to free it. The key strategy is always to isolate the subject on one side, then simplify.

What distinguishes harder rearrangements from basic ones?

At KS3 you learned to rearrange by doing the same operation to both sides — adding, subtracting, multiplying or dividing. Harder GCSE rearrangements introduce three new challenges:

Master each technique separately, then combine them for multi-step questions.

How do you rearrange when the subject is under a square root?

The inverse of square root is squaring. Square both sides first to remove the root, then isolate the subject.

Worked example 1: Make x the subject of y = √(3x − 1)

1. Square both sides: y² = 3x − 1
2. Add 1 to both sides: y² + 1 = 3x
3. Divide both sides by 3: x = (y² + 1) / 3

Check: Substitute x = (y² + 1)/3 back into 3x − 1: 3 × (y² + 1)/3 − 1 = y² + 1 − 1 = y². Then √(y²) = y. ✓

Remember: squaring both sides is only valid when both sides are non-negative. At GCSE this is assumed unless stated otherwise.

How do you rearrange when the subject is squared?

The inverse of squaring is taking the square root. Take the square root of both sides, and include a ± sign unless the context makes one sign impossible.

Worked example 2: Make r the subject of V = (4/3)πr³ ... actually, let's use A = πr²

Make r the subject of A = πr²:

1. Divide both sides by π: A/π = r²
2. Square root both sides: r = √(A/π)
3. Since r is a length (positive): r = √(A/π) (taking the positive root only)

Worked example 3: Make t the subject of s = ut + ½at²

This is more complex. Rearrange to isolate the t² term first:

1. Subtract ut from both sides: s − ut = ½at²
2. Multiply both sides by 2: 2(s − ut) = at²
3. Divide both sides by a: (2(s − ut))/a = t²
4. Square root both sides: t = √(2(s − ut)/a)

How do you rearrange when the subject appears twice?

When the subject (say x) appears in two different terms, you need to:

1. Multiply out any fractions to clear denominators.
2. Collect all terms containing x on one side.
3. Factorise x out of those terms.
4. Divide to isolate x.

Worked example 4: Make x the subject of y = (x + 2)/(x − 3)

1. Multiply both sides by (x − 3): y(x − 3) = x + 2
2. Expand the left side: yx − 3y = x + 2
3. Move all x terms to the left: yx − x = 2 + 3y
4. Factorise: x(y − 1) = 2 + 3y
5. Divide by (y − 1): x = (2 + 3y)/(y − 1)

The crucial insight is step 4: factorising x out of the bracket is what frees it from appearing twice.

How do you rearrange a formula involving a fraction with the subject in the denominator?

If x is in the denominator, multiply both sides by the denominator to move x to the numerator before doing anything else.

Worked example 5: Make t the subject of f = 1/t

1. Multiply both sides by t: ft = 1
2. Divide both sides by f: t = 1/f

Worked example 6: Make x the subject of p = 5/(x − 2)

1. Multiply both sides by (x − 2): p(x − 2) = 5
2. Expand: px − 2p = 5
3. Add 2p: px = 5 + 2p
4. Divide by p: x = (5 + 2p)/p

What is a good order of operations when tackling a harder rearrangement?

Follow this checklist:

1. Clear fractions first — multiply every term by the denominator.
2. Expand any brackets containing the subject.
3. Collect all subject-terms onto one side.
4. Factorise the subject out if it appears more than once.
5. Divide to leave the subject alone.
6. Square or square-root as needed for powers and roots.

Work step by step and write each line clearly. Missing a step causes cascading errors.

Frequently asked questions

How do I know whether to square or take a square root?

If the subject is under a square root (e.g. y = √x), you square both sides to remove the root. If the subject is squared (e.g. y = x²), you take the square root of both sides to free the subject. Think of it as applying the inverse operation to undo what is trapping the subject.

Must I always include ± when taking a square root?

In pure algebra, yes — x² = 9 gives x = ±3. But in physical contexts (where x might be a length or time) only the positive root is meaningful, and you write just the positive value. Follow any instructions in the question about the domain of the variable.

Why do I need to factorise when the subject appears twice?

Because you cannot divide by x if x appears in two separate terms — you need to group them first. Factorising creates a single bracket, after which division isolates the subject cleanly. If you forget to factorise and try to cancel x from one term only, you will get a wrong answer.

What if I cannot spot the right first step?

Write out every term that contains the subject, then ask "what is the biggest obstacle keeping the subject trapped?" Common obstacles are: fractions (multiply out), roots (square or cube), brackets (expand). Clear one obstacle at a time in the order that creates the least new complexity.

For Higher GCSE algebra tutoring with Professor Pi — visit aitutors.me.