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

The given equation represents an impossible relation,

...unless (see below)

If we attempt to simplify the given equation (by multiplying both sides by

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As an alternative answer to the one given above, it is possible to claim a pair of solutions:

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

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To simplify the equation x^2/(x^2-4) = x/(x+2)-2/(2-x), we can start by finding a common denominator for the fractions on the right side of the equation. The common denominator is (x+2)(2-x).

Next, we can rewrite the fractions on the right side with the common denominator:

x/(x+2) = (x(2-x))/((x+2)(2-x)) -2/(2-x) = (-2(x+2))/((x+2)(2-x))

Now, we can substitute these values back into the original equation:

x^2/(x^2-4) = (x(2-x))/((x+2)(2-x)) - (-2(x+2))/((x+2)(2-x))

Simplifying further, we can cancel out the common factors:

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

Expanding and simplifying the equation:

x^2 = 2x - x^2 + 2x + 4

Combining like terms:

2x^2 - 4x - 4 = 0

Factoring the quadratic equation:

2(x-2)(x+1) = 0

Setting each factor equal to zero:

x-2 = 0 or x+1 = 0

Solving for x:

x = 2 or x = -1

Therefore, the simplified solution to the equation x^2/(x^2-4) = x/(x+2)-2/(2-x) is x = 2 or x = -1.

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