# How do you solve x / 3 - 2 / 3 = 1 / x and find any extraneous solutions?

Feb 19, 2017

$x = 3 \text{ }$ or $\text{ } x = - 1$

#### Explanation:

Given:

$\frac{x}{3} - \frac{2}{3} = \frac{1}{x}$

Multiply through by $3 x$ to get:

${x}^{2} - 2 x = 3$

Note that this can only introduce an extraneous solution if the resulting equation has $x = 0$ as a root.

Subtract $3$ from both sides to get:

$0 = {x}^{2} - 2 x - 3 = \left(x - 3\right) \left(x + 1\right)$

Hence:

$x = 3 \text{ }$ or $\text{ } x = - 1$

These are both valid solutions of the original equation.