n(n+1)=380

Solution for n(n+1)=380 equation:

n(n+1)=380

We move all terms to the left:

n(n+1)-(380)=0

We multiply parentheses

n^2+n-380=0

a = 1; b = 1; c = -380;
Δ = b2-4ac
Δ = 12-4·1·(-380)
Δ = 1521
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1521}=39$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-39}{2*1}=\frac{-40}{2}
=-20
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+39}{2*1}=\frac{38}{2}
=19
$