x^2-6x+83=106

Solution for x^2-6x+83=106 equation:

x^2-6x+83=106

We move all terms to the left:

x^2-6x+83-(106)=0

We add all the numbers together, and all the variables

x^2-6x-23=0

a = 1; b = -6; c = -23;
Δ = b2-4ac
Δ = -62-4·1·(-23)
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{-(-6)-8\sqrt{2}}{2*1}=\frac{6-8\sqrt{2}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+8\sqrt{2}}{2*1}=\frac{6+8\sqrt{2}}{2}
$