Answer
The solutions are $x=\pm\sqrt{5}$ and $x=\pm2\sqrt{2}$
Work Step by Step
$x^{4}-13x^{2}+40=0$
Apply a variable change to solve the equation. First, let $u^{2}$ be equal to $x^{4}$:
$u^{2}=x^{4}$
$u=\sqrt{x^{4}}=x^{2}$
Substitute $x^{4}$ by $u^{2}$ and $x^{2}$ by $u$ in the original equation:
$u^{2}-13u+40=0$
Solve this equation by factoring:
$(u-5)(u-8)=0$
Set both factors equal to $0$ and solve each individual equation for $u$:
$u-5=0$
$u=5$
$u-8=0$
$u=8$
Substitute $u$ back to $x^{2}$ and solve for $x$:
$u=5$
$x^{2}=5$
$x=\pm\sqrt{5}$
$u=8$
$x^{2}=8$
$x=\pm2\sqrt{2}$
The solutions are $x=\pm\sqrt{5}$ and $x=\pm2\sqrt{2}$
