2*n(n-1)/2=132

Solution for 2*n(n-1)/2=132 equation:

2n(n-1)/2=132

We move all terms to the left:

2n(n-1)/2-(132)=0

We multiply all the terms by the denominator


2n(n-1)-132*2=0

We add all the numbers together, and all the variables

2n(n-1)-264=0

We multiply parentheses

2n^2-2n-264=0

a = 2; b = -2; c = -264;
Δ = b2-4ac
Δ = -22-4·2·(-264)
Δ = 2116
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{2116}=46$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-46}{2*2}=\frac{-44}{4}
=-11
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+46}{2*2}=\frac{48}{4}
=12
$