y^2=8y+1=0

Solution for y^2=8y+1=0 equation:

y^2=8y+1=0

We move all terms to the left:

y^2-(8y+1)=0

We get rid of parentheses

y^2-8y-1=0

a = 1; b = -8; c = -1;
Δ = b2-4ac
Δ = -82-4·1·(-1)
Δ = 68
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{68}=\sqrt{4*17}=\sqrt{4}*\sqrt{17}=2\sqrt{17}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{17}}{2*1}=\frac{8-2\sqrt{17}}{2}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{17}}{2*1}=\frac{8+2\sqrt{17}}{2}
$