k(k-1)=90

Solution for k(k-1)=90 equation:

k(k-1)=90

We move all terms to the left:

k(k-1)-(90)=0

We multiply parentheses

k^2-1k-90=0

a = 1; b = -1; c = -90;
Δ = b2-4ac
Δ = -12-4·1·(-90)
Δ = 361
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{361}=19$
$k_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-19}{2*1}=\frac{-18}{2}
=-9
$
$k_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+19}{2*1}=\frac{20}{2}
=10
$