S(x)=x^2-6x+1

Solution for S(x)=x^2-6x+1 equation:

(S)=S^2-6S+1

We move all terms to the left:

(S)-(S^2-6S+1)=0

We get rid of parentheses

-S^2+S+6S-1=0

We add all the numbers together, and all the variables

-1S^2+7S-1=0

a = -1; b = 7; c = -1;
Δ = b2-4ac
Δ = 72-4·(-1)·(-1)
Δ = 45
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{45}=\sqrt{9*5}=\sqrt{9}*\sqrt{5}=3\sqrt{5}$
$S_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(7)-3\sqrt{5}}{2*-1}=\frac{-7-3\sqrt{5}}{-2}
$
$S_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(7)+3\sqrt{5}}{2*-1}=\frac{-7+3\sqrt{5}}{-2}
$