# How do you solve x/(x+2) - 2/(x-2) = (x^2+4)/(x^2-4)?

May 2, 2016

Multiply through by $\left(x - 2\right) \left(x + 2\right)$ and simplify to attempt to find a solution, but the only possibility turns out to be a spurious solution.

#### Explanation:

Given:

$\frac{x}{x + 2} - \frac{2}{x - 2} = \frac{{x}^{2} + 4}{{x}^{2} - 4}$

Note that ${x}^{2} - 4 = \left(x - 2\right) \left(x + 2\right)$

Multiply both sides by $\left(x - 2\right) \left(x + 2\right)$ to get:

$x \left(x - 2\right) - 2 \left(x + 2\right) = {x}^{2} + 4$

Note that this potentially (and does) introduces spurious solutions for $x = \pm 2$

The left hand side simplifies as follows:

$x \left(x - 2\right) - 2 \left(x + 2\right) = {x}^{2} - 2 x - 2 x - 4 = {x}^{2} - 4 x - 4$

So the equation becomes:

$\textcolor{red}{\cancel{\textcolor{b l a c k}{{x}^{2}}}} - 4 x - 4 = \textcolor{red}{\cancel{\textcolor{b l a c k}{{x}^{2}}}} + 4$

Subtract ${x}^{2}$ from both sides and add $4$ to both sides to get:

$- 4 x = 8$

Divide both sides by $- 4$ to get:

$x = - 2$

This is not a solution of the original equation, since if $x = - 2$ then the denominators of two of the expressions are zero.

So the original problem has no solutions.