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

Apr 14, 2017

The given equation represents an impossible relation,
...unless (see below)

#### Explanation:

If we attempt to simplify the given equation (by multiplying both sides by $\left({x}^{2} - 4\right)$ with the assumption $x \notin \left\{- 2 , + 2\right\}$

x^2=x(x-2)+(2(x+2)

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

$0 = 4 \textcolor{w h i t e}{\text{XXXX}}$Impossible!

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As an alternative answer to the one given above, it is possible to claim a pair of solutions: $\textcolor{m a \ge n t a}{x = - 2}$ or $\textcolor{m a \ge n t a}{x = + 2}$

and if we look at the graphs for the left and right sides of the given equation this (sort of) makes sense:

The limits at $x = \pm 2$ when approached from the same sides are equal.