k(k+1)-66.6=0

Solution for k(k+1)-66.6=0 equation:

k(k+1)-66.6=0

We multiply parentheses

k^2+k-66.6=0

a = 1; b = 1; c = -66.6;
Δ = b2-4ac
Δ = 12-4·1·(-66.6)
Δ = 267.4
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-\sqrt{267.4}}{2*1}=\frac{-1-\sqrt{267.4}}{2}
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+\sqrt{267.4}}{2*1}=\frac{-1+\sqrt{267.4}}{2}
$