Question

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

Answer 1

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

Explanation:

If we attempt to simplify the given equation (by multiplying both sides by `(x^2-4)` with the assumption `x !in {-2,+2}`

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

`color(red)(cancel(color(black)(x^2)))=color(red)(cancel(color(black)(x^2)))color(blue)(cancel(color(black)(-2x)))color(blue)(cancel(color(black)(+2x)))+4`

`0=4color(white)("XXXX")`Impossible!

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As an alternative answer to the one given above, it is possible to claim a pair of solutions: `color(magenta)(x=-2)` or `color(magenta)(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=+-2` when approached from the same sides are equal.