Harder rearranging: when the subject appears twice

When the letter you want appears in two places, gather both onto one side and factorise it out.

Practise this lesson free (6 questions)

The method

1. Clear any fractions or roots.
2. Expand brackets.
3. Collect every term containing the subject on one side, everything else on the other.
4. Factorise out the subject.
5. Divide by the bracket.

Example

Make x the subject of y = (x + 2)(x − 3)
Multiply by (x − 3): y(x − 3) = x + 2
Expand: xy − 3y = x + 2
Collect x terms: xy − x = 3y + 2
Factorise: x(y − 1) = 3y + 2
Divide: x = (3y + 2)(y − 1)

Powers and roots

Make x the subject of y = √((x + a)b)
Square both sides: y2 = (x + a)b
Multiply by b: by2 = x + a
x = by2 − a
Do the opposite operations in the reverse order.
Worked example

Make x the subject of 3x + a = bx + 5.

  1. Collect x terms on the left: 3x − bx = 5 − a
  2. Factorise: x(3 − b) = 5 − a
  3. Divide: x = (5 − a)(3 − b)

Answer: x = (5 − a)(3 − b)

Key idea

Get every subject term on one side, factorise the subject out, then divide by the bracket.

Key words
Factorise
Write an expression as a product by taking out common factors or putting it into brackets.
Term
A single number, letter or product of them in an expression, such as 5x or −3.

All GCSE and International GCSE Mathematics key terms and definitions →

Quick check

Try two quick questions

See if it has sunk in. Answers are marked straight away.

Extra practice

2 more questions

Try each one, then tap to check.

  1. y = (x + 4)(x − 6). Find x when y = 2.

    Show the answer

    Answer: 16

    2(x − 6) = x + 4, so 2x − 12 = x + 4 and x = 16.

  2. In ax + 3 = bx + 7, what is the first step to make x the subject?

    1. Factorise the 7
    2. Divide both sides by a
    3. Subtract 3 from the left side only
    4. Collect the x terms on one side: ax − bx = 4
    Show the answer

    Answer: Collect the x terms on one side: ax − bx = 4

    Get all x terms together, then factorise and divide.

Check you have got it

Answer 6 quick questions with instant marking. If you get one wrong, GCSE-ready shows you why and gives you another go. It is free, and you do not need an account.

Practise this lesson free (6 questions)All GCSE and International GCSE Mathematics lessons

Video for this lesson coming soon. Subscribe on YouTube

Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

Related lessons