31-2x=x^2

Solution for 31-2x=x^2 equation:

31-2x=x^2

We move all terms to the left:

31-2x-(x^2)=0

determiningTheFunctionDomain
-x^2-2x+31=0

We add all the numbers together, and all the variables

-1x^2-2x+31=0

a = -1; b = -2; c = +31;
Δ = b2-4ac
Δ = -22-4·(-1)·31
Δ = 128
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{128}=\sqrt{64*2}=\sqrt{64}*\sqrt{2}=8\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-8\sqrt{2}}{2*-1}=\frac{2-8\sqrt{2}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+8\sqrt{2}}{2*-1}=\frac{2+8\sqrt{2}}{-2}
$