n(n-1)/2=231

Solution for n(n-1)/2=231 equation:

n(n-1)/2=231

We move all terms to the left:

n(n-1)/2-(231)=0

We multiply all the terms by the denominator


n(n-1)-231*2=0

We add all the numbers together, and all the variables

n(n-1)-462=0

We multiply parentheses

n^2-1n-462=0

a = 1; b = -1; c = -462;
Δ = b2-4ac
Δ = -12-4·1·(-462)
Δ = 1849
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1849}=43$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-43}{2*1}=\frac{-42}{2}
=-21
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+43}{2*1}=\frac{44}{2}
=22
$