Undo in reverse, same to both sides
Whatever is done to the subject (×, +, square, …) is undone in reverse order, applied to the whole of both sides to keep the equation balanced.
Subject appears twice: collect and factorise
Gather every term containing the subject on one side, factorise it out, then divide. e.g. from you get .
A square undoes a square root, and vice versa
To free a squared subject, square root both sides; to free a square-rooted subject, square both sides — square the entire side, not term by term.
Drawn from real examiner reports.
Not factorising a twice-appearing subject
Students leave the subject in two places and cannot finish; the key step of factorising it out to isolate it is missed.
A commonly reported examiner note records this on a recent higher-tier paper.
Sign errors and mishandling squared terms
When squaring both sides or moving terms across, students drop or flip signs, or square only one term of a side.
A commonly reported examiner note records this on a recent higher-tier paper.
Trying to cancel across a sum
Cancelling a letter that is added (not a factor of the whole side) — only common factors of the entire side may be cancelled.
Squaring before isolating the root
To free a subject inside a square root you must get the root alone first, then square both sides. Squaring while another factor still multiplies the root — as in — leaves the root in place and makes the algebra messier.
Adding ± to a cube root
A cube root gives a single real value: only, never . The belongs to square roots, where both signs can satisfy the equation. Watch this when a subject sits inside a cube such as a sphere-volume formula.
Sanity-check with a number
Substitute simple numbers into the original formula and into your rearrangement; they should give the same value.
Rearrange first, then substitute numbers
When a formula gives values for every letter except one, make that letter the subject before putting the numbers in. Rearranging symbolically first keeps the working clean and avoids arithmetic clutter that hides slips.
Isolate the root before squaring
Get the square root on its own on one side, then square both sides so the root disappears cleanly. In , divide by first to reach .
Changing the subject is just solving for a letter. Undo the operations in reverse order, doing the same to both sides.
Example: make the subject of .
Golden rule of rearranging?
Make the subject of .
