x(x+1)=13

Solution for x(x+1)=13 equation:

x(x+1)=13

We move all terms to the left:

x(x+1)-(13)=0

We multiply parentheses

x^2+x-13=0

a = 1; b = 1; c = -13;
Δ = b2-4ac
Δ = 12-4·1·(-13)
Δ = 53
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-\sqrt{53}}{2*1}=\frac{-1-\sqrt{53}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+\sqrt{53}}{2*1}=\frac{-1+\sqrt{53}}{2}
$