Y=x^2+12x+31

Solution for Y=x^2+12x+31 equation:

=Y^2+12Y+31

We move all terms to the left:

-(Y^2+12Y+31)=0

We get rid of parentheses

-Y^2-12Y-31=0

We add all the numbers together, and all the variables

-1Y^2-12Y-31=0

a = -1; b = -12; c = -31;
Δ = b2-4ac
Δ = -122-4·(-1)·(-31)
Δ = 20
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:

$\sqrt{\Delta}=\sqrt{20}=\sqrt{4*5}=\sqrt{4}*\sqrt{5}=2\sqrt{5}$
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{5}}{2*-1}=\frac{12-2\sqrt{5}}{-2}
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{5}}{2*-1}=\frac{12+2\sqrt{5}}{-2}
$