Answer
$x=1$
Work Step by Step
We are given:
$-x^{-2}+2x^{-1}=1$
$-x^{-2}+2x^{-1}-1=0$
We multiply through by -1:
$-1(-x^{-2}+2x^{-1}-1)=-1(0)$
$x^{-2}-2x^{-1}+1=0$
We multiply through again by $x^2$:
$x^2(x^{-2}-2x^{-1}+1)=x^2(0)$
$1-2x+x^2=0$
$x^2-2x+1=0$
And factor:
$(x-1)(x-1)=0$
$(x-1)^2=0$
Equate each factor to zero, tehn solve each equation for $x$:
$x-1=0$
$x=1$