n*(n-1)/2=66

Solution for n*(n-1)/2=66 equation:

n(n-1)/2=66

We move all terms to the left:

n(n-1)/2-(66)=0

We multiply all the terms by the denominator


n(n-1)-66*2=0

We add all the numbers together, and all the variables

n(n-1)-132=0

We multiply parentheses

n^2-1n-132=0

a = 1; b = -1; c = -132;
Δ = b2-4ac
Δ = -12-4·1·(-132)
Δ = 529
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{529}=23$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-23}{2*1}=\frac{-22}{2}
=-11
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+23}{2*1}=\frac{24}{2}
=12
$