209=x^2+x

Solution for 209=x^2+x equation:

209=x^2+x

We move all terms to the left:

209-(x^2+x)=0

We get rid of parentheses

-x^2-x+209=0

We add all the numbers together, and all the variables

-1x^2-1x+209=0

a = -1; b = -1; c = +209;
Δ = b2-4ac
Δ = -12-4·(-1)·209
Δ = 837
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:

$\sqrt{\Delta}=\sqrt{837}=\sqrt{9*93}=\sqrt{9}*\sqrt{93}=3\sqrt{93}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-3\sqrt{93}}{2*-1}=\frac{1-3\sqrt{93}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+3\sqrt{93}}{2*-1}=\frac{1+3\sqrt{93}}{-2}
$