n(n-2)/2=78

Solution for n(n-2)/2=78 equation:

n(n-2)/2=78

We move all terms to the left:

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

n(n-2)-156=0

We multiply parentheses

n^2-2n-156=0

a = 1; b = -2; c = -156;
Δ = b2-4ac
Δ = -22-4·1·(-156)
Δ = 628
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{628}=\sqrt{4*157}=\sqrt{4}*\sqrt{157}=2\sqrt{157}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{157}}{2*1}=\frac{2-2\sqrt{157}}{2}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{157}}{2*1}=\frac{2+2\sqrt{157}}{2}
$