Answer
$0$.
Work Step by Step
The given expression is
$\Rightarrow \frac{x-4}{x^2-4}=\frac{-2}{x-2}$
Use the cross products property.
$\Rightarrow (x-4)(x-2)=(x^2-4)(-2)$
Simplify.
$\Rightarrow x^2-6x+8=-2x^2+8$
Arrange terms on one side.
$\Rightarrow x^2-6x+8+2x^2-8=0$
Add like terms.
$\Rightarrow 3x^2-6x=0$
Factor out $3x$.
$\Rightarrow 3x(x-2)=0$
Use zero product property.
$3x=0$ or $x-2=0$
Solve for $x$.
$x=0$ or $x=2$
Check for $x=0$.
$\Rightarrow \frac{0-4}{0^2-4}=\frac{-2}{0-2}$
$\Rightarrow \frac{-4}{-4}=\frac{-2}{-2}$
$\Rightarrow 1=1$
Check for $x=2$.
$\Rightarrow \frac{2-4}{2^2-4}=\frac{-2}{2-2}$
$\Rightarrow \frac{-2}{4-4}=\frac{-2}{0}$
$\Rightarrow \frac{-2}{0}=\frac{-2}{0}$
Undefined.
Hence, the solution is $x=0$.
