x*x+x=61

Solution for x*x+x=61 equation:

x*x+x=61

We move all terms to the left:

x*x+x-(61)=0

We add all the numbers together, and all the variables

x+x*x-61=0

Wy multiply elements

x^2+x-61=0

a = 1; b = 1; c = -61;
Δ = b2-4ac
Δ = 12-4·1·(-61)
Δ = 245
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:

$\sqrt{\Delta}=\sqrt{245}=\sqrt{49*5}=\sqrt{49}*\sqrt{5}=7\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-7\sqrt{5}}{2*1}=\frac{-1-7\sqrt{5}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+7\sqrt{5}}{2*1}=\frac{-1+7\sqrt{5}}{2}
$