x+x^2+1=12

Solution for x+x^2+1=12 equation:

x+x^2+1=12

We move all terms to the left:

x+x^2+1-(12)=0

We add all the numbers together, and all the variables

x^2+x-11=0

a = 1; b = 1; c = -11;
Δ = b2-4ac
Δ = 12-4·1·(-11)
Δ = 45
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:

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