Answer
The solutions are $x=4$, $x=-2$, $x=3$ and $x=-1$
Work Step by Step
$(x^{2}-2x)^{2}-11(x^{2}-2x)+24=0$
Let $u$ be equal to $x^{2}-2x$
If $u=x^{2}-2x$, then $u^{2}=(x^{2}-2x)^{2}$
Rewrite the original equation using the new variable $u$:
$u^{2}-11u+24=0$
Solve by factoring:
$(u-8)(u-3)=0$
Set both factors equal to $0$ and solve each individual equation for $u$:
$u-8=0$
$u=8$
$u-3=0$
$u=3$
Substitute $u$ back to $x^{2}-2x$ and solve for $x$:
$x^{2}-2x=8$
$x^{2}-2x-8=0$
$(x-4)(x+2)=0$
$x-4=0$
$x=4$
$x+2=0$
$x=-2$
$x^{2}-2x=3$
$x^{2}-2x-3=0$
$(x-3)(x+1)=0$
$x-3=0$
$x=3$
$x+1=0$
$x=-1$
The solutions are $x=4$, $x=-2$, $x=3$ and $x=-1$
