2y(2y-3)+2=31

Solution for 2y(2y-3)+2=31 equation:

2y(2y-3)+2=31

We move all terms to the left:

2y(2y-3)+2-(31)=0

We add all the numbers together, and all the variables

2y(2y-3)-29=0

We multiply parentheses

4y^2-6y-29=0

a = 4; b = -6; c = -29;
Δ = b2-4ac
Δ = -62-4·4·(-29)
Δ = 500
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:

$\sqrt{\Delta}=\sqrt{500}=\sqrt{100*5}=\sqrt{100}*\sqrt{5}=10\sqrt{5}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-10\sqrt{5}}{2*4}=\frac{6-10\sqrt{5}}{8}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+10\sqrt{5}}{2*4}=\frac{6+10\sqrt{5}}{8}
$