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

1 Answer
Apr 14, 2017

Answer:

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:
enter image source here
The limits at #x=+-2# when approached from the same sides are equal.