Answer
The solutions are $x=\dfrac{1}{5}$ and $x=-\dfrac{1}{4}$
Work Step by Step
$x^{-2}-x^{-1}-20=0$
Let $u$ be equal to $x^{-1}$
If $u=x^{-1}$, then $u^{2}=x^{-2}$
Rewrite the original equation using the new variable $u$:
$u^{2}-u-20=0$
Solve by factoring:
$(u-5)(u+4)=0$
Set both factors equal to $0$ and solve each individual equation for $u$:
$u-5=0$
$u=5$
$u+4=0$
$u=-4$
Substitute $u$ back to $x^{-1}$ and solve for $x$:
$x^{-1}=5$
$\dfrac{1}{x}=5$
$x=\dfrac{1}{5}$
$x^{-1}=-4$
$\dfrac{1}{x}=-4$
$x=-\dfrac{1}{4}$
The solutions are $x=\dfrac{1}{5}$ and $x=-\dfrac{1}{4}$
