6=6x+x^2

Solution for 6=6x+x^2 equation:

6=6x+x^2

We move all terms to the left:

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

We get rid of parentheses

-x^2-6x+6=0

We add all the numbers together, and all the variables

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

a = -1; b = -6; c = +6;
Δ = b2-4ac
Δ = -62-4·(-1)·6
Δ = 60
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:

$\sqrt{\Delta}=\sqrt{60}=\sqrt{4*15}=\sqrt{4}*\sqrt{15}=2\sqrt{15}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{15}}{2*-1}=\frac{6-2\sqrt{15}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{15}}{2*-1}=\frac{6+2\sqrt{15}}{-2}
$