Answer
The solutions are $y=-8$, $y=1$, $y=4$ and $y=-2$
Work Step by Step
$\Big(y-\dfrac{8}{y}\Big)^{2}+5\Big(y-\dfrac{8}{y}\Big)-14=0$
Let $x$ be equal to $y-\dfrac{8}{y}$
If $x=y-\dfrac{8}{y}$, then $x^{2}=\Big(y-\dfrac{8}{y}\Big)^{2}$
Rewrite the original equation using the new variable $x$:
$x^{2}+5x-14=0$
Solve by factoring:
$(x+7)(x-2)=0$
Set both factors equal to $0$ and solve each individual equation for $x$:
$x+7=0$
$x=-7$
$x-2=0$
$x=2$
Substitute $x$ back to $y-\dfrac{8}{y}$ and solve for $y$:
$y-\dfrac{8}{y}=-7$
$\dfrac{y^{2}-8}{y}=-7$
$y^{2}-8=-7y$
$y^{2}+7y-8=0$
$(y+8)(y-1)=0$
$y+8=0$
$y=-8$
$y-1=0$
$y=1$
$y-\dfrac{8}{y}=2$
$\dfrac{y^{2}-8}{y}=2$
$y^{2}-8=2y$
$y^{2}-2y-8=0$
$(y-4)(y+2)=0$
$y-4=0$
$y=4$
$y+2=0$
$y=-2$
The solutions are $y=-8$, $y=1$, $y=4$ and $y=-2$
