n(n+1)/2=78

Solution for n(n+1)/2=78 equation:

n(n+1)/2=78

We move all terms to the left:

n(n+1)/2-(78)=0

We multiply all the terms by the denominator


n(n+1)-78*2=0

We add all the numbers together, and all the variables

n(n+1)-156=0

We multiply parentheses

n^2+n-156=0

a = 1; b = 1; c = -156;
Δ = b2-4ac
Δ = 12-4·1·(-156)
Δ = 625
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{625}=25$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-25}{2*1}=\frac{-26}{2}
=-13
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+25}{2*1}=\frac{24}{2}
=12
$