Answer
The solutions are $x=2$, $x=7$ and $x=-5$
Work Step by Step
$x(x-2)^{3}-35(x-2)^{2}=0$
Take out common factor $(x-2)^{2}$:
$(x-2)^{2}[x(x-2)-35]=0$
$(x-2)^{2}(x^{2}-2x-35)=0$
Factor $x^{2}-2x-35$:
$(x-2)^{2}(x-7)(x+5)=0$
Set all three factors equal to $0$ and solve each individual equation for $x$:
$(x-2)^{2}=0$
$\sqrt{(x-2)^{2}}=\sqrt{0}$
$x-2=0$
$x=2$
$x-7=0$
$x=7$
$x+5=0$
$x=-5$
The solutions are $x=2$, $x=7$ and $x=-5$
