y^2-12y=31

Solution for y^2-12y=31 equation:

y^2-12y=31

We move all terms to the left:

y^2-12y-(31)=0

a = 1; b = -12; c = -31;
Δ = b2-4ac
Δ = -122-4·1·(-31)
Δ = 268
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:

$\sqrt{\Delta}=\sqrt{268}=\sqrt{4*67}=\sqrt{4}*\sqrt{67}=2\sqrt{67}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{67}}{2*1}=\frac{12-2\sqrt{67}}{2}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{67}}{2*1}=\frac{12+2\sqrt{67}}{2}
$