Fractional equation with factorable denominators

Text

Solve the following equation: $\frac{x}{x^2 - 4} = \frac{2}{x + 2}.$

Solution

Factor $x^2-4=(x-2)(x+2)$; existence conditions $x\neq \pm 2$. Multiplying by $(x-2)(x+2)$: $x = 2(x-2) = 2x - 4 \;\Rightarrow\; -x = -4 \;\Rightarrow\; x = 4.$ The value $x=4$ is acceptable (different from $\pm 2$). $\boxed{x = 4}$