Solve x² = 5x: Why Dividing by x Loses an Answer

The question

Solve $x^2 = 5x$ .

The common wrong answer

Most students divide both sides by $x$ :

x^2 = 5x \quad\Rightarrow\quad x = 5

This looks fine, but it’s only half the answer. Dividing by $x$ quietly assumes $x$ isn’t 0, and $x = 0$ is also a solution: $0^2 = 5 \times 0$ .

The correct working

Bring everything to one side and factorise:

x^2 - 5x = 0 \quad\Rightarrow\quad x(x - 5) = 0

So $x = 0$ or $x = 5$ .

Why it works

Factorising works because of one simple rule: if two things multiply to give zero, at least one of them must be zero.

x(x - 5) = 0 \quad\Rightarrow\quad x = 0 \quad\text{or}\quad x - 5 = 0

Dividing both sides by $x$ is only allowed when $x \ne 0$ , because you can’t divide by zero. So the moment you divide by $x$ , you have silently assumed $x$ isn’t 0, and the solution $x = 0$ disappears from your working. Keeping everything on one side and factorising never throws a solution away.

How to spot it next time

Never divide both sides by something that contains the unknown. Move everything to one side and factorise instead.
A quadratic equation can have up to two solutions. If you’ve only found one, check whether you’ve lost the other.
The same trap appears in trigonometry: in $\sin x \cos x = \sin x$ , dividing by $\sin x$ leaves $\cos x = 1$ , which loses solutions such as $x = 180^\circ$ , where $\sin x = 0$ .