n(n-1)/2=120

Solution for n(n-1)/2=120 equation:

n(n-1)/2=120

We move all terms to the left:

n(n-1)/2-(120)=0

We multiply all the terms by the denominator


n(n-1)-120*2=0

We add all the numbers together, and all the variables

n(n-1)-240=0

We multiply parentheses

n^2-1n-240=0

a = 1; b = -1; c = -240;
Δ = b2-4ac
Δ = -12-4·1·(-240)
Δ = 961
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{961}=31$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-31}{2*1}=\frac{-30}{2}
=-15
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+31}{2*1}=\frac{32}{2}
=16
$