n(n-3)/2=90

Solution for n(n-3)/2=90 equation:

n(n-3)/2=90

We move all terms to the left:

n(n-3)/2-(90)=0

We multiply all the terms by the denominator


n(n-3)-90*2=0

We add all the numbers together, and all the variables

n(n-3)-180=0

We multiply parentheses

n^2-3n-180=0

a = 1; b = -3; c = -180;
Δ = b2-4ac
Δ = -32-4·1·(-180)
Δ = 729
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{729}=27$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-27}{2*1}=\frac{-24}{2}
=-12
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+27}{2*1}=\frac{30}{2}
=15
$