x+239=x^2+111

Solution for x+239=x^2+111 equation:

x+239=x^2+111

We move all terms to the left:

x+239-(x^2+111)=0

We get rid of parentheses

-x^2+x-111+239=0

We add all the numbers together, and all the variables

-1x^2+x+128=0

a = -1; b = 1; c = +128;
Δ = b2-4ac
Δ = 12-4·(-1)·128
Δ = 513
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:

$\sqrt{\Delta}=\sqrt{513}=\sqrt{9*57}=\sqrt{9}*\sqrt{57}=3\sqrt{57}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-3\sqrt{57}}{2*-1}=\frac{-1-3\sqrt{57}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+3\sqrt{57}}{2*-1}=\frac{-1+3\sqrt{57}}{-2}
$