Question

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

Answer 1

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

Explanation:

Given:

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

Note that `x^2-4 = (x-2)(x+2)`

Multiply both sides by `(x-2)(x+2)` to get:

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

Note that this potentially (and does) introduces spurious solutions for `x = +-2`

The left hand side simplifies as follows:

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

So the equation becomes:

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

Subtract `x^2` from both sides and add `4` to both sides to get:

`-4x = 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.