2x^2-12x+31=x^2

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

2x^2-12x+31=x^2

We move all terms to the left:

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

determiningTheFunctionDomain
2x^2-x^2-12x+31=0

We add all the numbers together, and all the variables

x^2-12x+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:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

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