Answer
$x=\frac{3}{4}$ or $x=\frac{4}{3}$
Work Step by Step
$\displaystyle \frac{1}{(x-1)^{2}}+\frac{1}{x-1}=12$
We multiply both sides by $(x-1)^2$:
$1+(x-1)=12(x-1)^2$
$12(x-1)^2-(x-1)-1=0$
$[4(x-1)+1][3(x-1)-1]=0$
$(4x-4+1)(3x-3-1)=0$
$(4x-3)(3x-4)=0$
$(4x-3)=0$ or $(3x-4)=0$
$x=\frac{3}{4}$ or $x=\frac{4}{3}$