w(w+1)=30

Solution for w(w+1)=30 equation:

w(w+1)=30

We move all terms to the left:

w(w+1)-(30)=0

We multiply parentheses

w^2+w-30=0

a = 1; b = 1; c = -30;
Δ = b2-4ac
Δ = 12-4·1·(-30)
Δ = 121
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$w_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$w_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{121}=11$
$w_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-11}{2*1}=\frac{-12}{2}
=-6
$
$w_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+11}{2*1}=\frac{10}{2}
=5
$