X(x+1)=137

Solution for X(x+1)=137 equation:

X(X+1)=137

We move all terms to the left:

X(X+1)-(137)=0

We multiply parentheses

X^2+X-137=0

a = 1; b = 1; c = -137;
Δ = b2-4ac
Δ = 12-4·1·(-137)
Δ = 549
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:

$\sqrt{\Delta}=\sqrt{549}=\sqrt{9*61}=\sqrt{9}*\sqrt{61}=3\sqrt{61}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-3\sqrt{61}}{2*1}=\frac{-1-3\sqrt{61}}{2}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+3\sqrt{61}}{2*1}=\frac{-1+3\sqrt{61}}{2}
$