Rearranging Complex Formulas

Rearranging harder formulae: undoing powers and roots, and making a subject that appears twice the subject by collecting and factorising.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024, Edexcel Further Pure Maths 4PM1 · 10 real past-paper questions below

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Undo in Reverse Order

To make a letter the subject, undo what was done to it, last thing first. A cube is undone by a cube root, a square by a square root.

Worked example

1
$y = 2 t^{3} + 5 k$ — Make t the subject. t was cubed, then doubled, then 5k was added. We undo those in reverse.
2
$y - 5 k = 2 t^{3}$ — Take 5k from both sides. The +5k was done last, so it is undone first.
3
$\frac{y - 5 k}{2} = t^{3}$ — Divide both sides by 2. The 2 divides the whole side, so y − 5k goes on top.
4
$t = 3 \frac{y - 5 k}{2}$ — Cube root both sides. The root covers the whole side, not just the y.

A square works the same way, with one extra thing to remember: a square root can be + or −.

Worked example

1
$A = k (r^{2} + h^{2})$ — Make r the subject. r was squared, then $h^{2}$ was added, then the bracket was multiplied by k.
2
$\frac{A}{k} = r^{2} + h^{2}$ — Divide both sides by k. The k multiplies the bracket, and it was done last.
3
$\frac{A}{k} - h^{2} = r^{2}$ — Take $h^{2}$ from both sides so $r^{2}$ is on its own.
4
$r = \pm \frac{A}{k} - h^{2}$ — Square root both sides. It covers both terms. Keep ± unless r is a length, which cannot be negative.

Common mistake

Rooting each term on its own. $9 + 16 = 5$ , but $9 + 16 = 7$ . The root goes over the whole side.

Subject Appears Twice (Extended/Higher)

The 4-mark question: the subject is on the top and the bottom of a fraction, under a root. Work through it one move at a time.

Worked example

1
$k = \frac{3 b + m}{b + 4}$ — Make b the subject. b is in two places, inside a fraction, inside a square root.
2
$k^{2} = \frac{3 b + m}{b + 4}$ — Square both sides. The root was done last, so it comes off first.
3
$k^{2} (b + 4) = 3 b + m$ — Multiply both sides by the bottom, $b + 4$ . Now there is no fraction.
4
$k^{2} b + 4 k^{2} = 3 b + m$ — Expand the bracket: $k^{2} \times b$ and $k^{2} \times 4$ . The b terms are marked.
5
$k^{2} b - 3 b = m - 4 k^{2}$ — Take 3b from both sides and take $4 k^{2}$ from both sides. All the b terms are now on the left.
6
$b (k^{2} - 3) = m - 4 k^{2}$ — Factorise: both terms on the left have b, so take b out. Now b appears once.
7
$b = \frac{m - 4 k ^{2}}{k ^{2} - 3}$ — Divide both sides by $k^{2} - 3$ to leave b on its own.

Key idea

When the subject appears twice, it is like two friends who will not leave a party separately. Gather them on one side, take them out as a factor, and they leave together.

Key idea

Show every line. In a 4-mark question you get a mark for clearing the root or fraction, for expanding, for collecting, and for factorising. A slip in the last line still keeps 3 marks; a bare answer that is wrong scores 0.

Signs When You Collect (Extended/Higher)

When there is a number in front of a bracket, or minus signs, the same method works. Your answer may look different from the mark scheme but be equal.

Worked example

1
$w = \frac{3 ( t - 2 )}{t + 4}$ — Make t the subject. t is on the top and the bottom.
2
$w (t + 4) = 3 (t - 2)$ — Multiply both sides by the bottom, $t + 4$ , to clear the fraction.
3
$w t + 4 w = 3 t - 6$ — Expand both brackets: $3 \times (- 2) = - 6$ . The t terms are marked.
4
$w t - 3 t = - 6 - 4 w$ — Take 3t from both sides and 4w from both sides. The −6 stays on the right with its minus.
5
$t (w - 3) = - 6 - 4 w$ — Factorise: take t out of the left side, so t appears once.
6
$t = \frac{- 6 - 4 w}{w - 3}$ — Divide both sides by $w - 3$ . This answer is correct.

Key idea

Times the top and bottom by −1 and you get $\frac{4 w + 6}{3 - w}$ . Both are right, and the second has no leading minus signs.

Which Method? Read the Question

Find the subject, then see what surrounds it.

You see	You do
A power on the subject	undo it last: root
A root over everything	square first
A fraction	multiply by the bottom
Subject twice	collect, then factorise
One step only	Rearranging Formulas

Key idea

Check with easy numbers. Put them into your answer, then back into the formula.

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