Answer
The solutions are $y=-10$, $y=1$, $y=5$ and $y=-2$
Work Step by Step
$\Big(y-\dfrac{10}{y}\Big)^{2}+6\Big(y-\dfrac{10}{y}\Big)-27=0$
Let $u$ be equal to $y-\dfrac{10}{y}$
If $u=y-\dfrac{10}{y}$, then $u^{2}=\Big(y-\dfrac{10}{y}\Big)^{2}$
Rewrite the original equation using the new variable $u$:
$u^{2}+6u-27=0$
Solve by factoring:
$(u+9)(u-3)=0$
Set both factors equal to $0$ and solve each individual equation for $u$:
$u+9=0$
$u=-9$
$u-3=0$
$u=3$
Substitute $u$ back to $y-\dfrac{10}{y}$ and solve for $y$:
$y-\dfrac{10}{y}=-9$
$\dfrac{y^{2}-10}{y}=-9$
$y^{2}-10=-9y$
$y^{2}+9y-10=0$
$(y+10)(y-1)=0$
$y+10=0$
$y=-10$
$y-1=0$
$y=1$
$y-\dfrac{10}{y}=3$
$\dfrac{y^{2}-10}{y}=3$
$y^{2}-10=3y$
$y^{2}-3y-10=0$
$(y-5)(y+2)=0$
$y-5=0$
$y=5$
$y+2=0$
$y=-2$
The solutions are $y=-10$, $y=1$, $y=5$ and $y=-2$
