x(x+1)=181

Solution for x(x+1)=181 equation:

x(x+1)=181

We move all terms to the left:

x(x+1)-(181)=0

We multiply parentheses

x^2+x-181=0

a = 1; b = 1; c = -181;
Δ = b2-4ac
Δ = 12-4·1·(-181)
Δ = 725
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:

$\sqrt{\Delta}=\sqrt{725}=\sqrt{25*29}=\sqrt{25}*\sqrt{29}=5\sqrt{29}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-5\sqrt{29}}{2*1}=\frac{-1-5\sqrt{29}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+5\sqrt{29}}{2*1}=\frac{-1+5\sqrt{29}}{2}
$