Answer
The solutions are $x=-\dfrac{1}{2}$ and $x=\dfrac{1}{3}$
Work Step by Step
$x^{-2}-x^{-1}-6=0$
Let $u$ be equal to $x^{-1}$
If $u=x^{-1}$, then $u^{2}=x^{-2}$
Rewrite the original equation using the new variable $u$:
$u^{2}-u-6=0$
Solve by factoring:
$(u-3)(u+2)=0$
Set both factors equal to $0$ and solve each individual equation for $u$:
$u-3=0$
$u=3$
$u+2=0$
$u=-2$
Substitute $u$ back to $x^{-1}$ and solve for $x$:
$x^{-1}=3$
$\dfrac{1}{x}=3$
$x=\dfrac{1}{3}$
$x^{-1}=-2$
$\dfrac{1}{x}=-2$
$x=-\dfrac{1}{2}$
The solutions are $x=-\dfrac{1}{2}$ and $x=\dfrac{1}{3}$
