n(n-3)/2=97

Solution for n(n-3)/2=97 equation:

n(n-3)/2=97

We move all terms to the left:

n(n-3)/2-(97)=0

We multiply all the terms by the denominator


n(n-3)-97*2=0

We add all the numbers together, and all the variables

n(n-3)-194=0

We multiply parentheses

n^2-3n-194=0

a = 1; b = -3; c = -194;
Δ = b2-4ac
Δ = -32-4·1·(-194)
Δ = 785
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-\sqrt{785}}{2*1}=\frac{3-\sqrt{785}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+\sqrt{785}}{2*1}=\frac{3+\sqrt{785}}{2}
$