n(n+1,)=182

Solution for n(n+1,)=182 equation:

n(n+1.)=182

We move all terms to the left:

n(n+1.)-(182)=0

We add all the numbers together, and all the variables

n(n+1)-182=0

We multiply parentheses

n^2+n-182=0

a = 1; b = 1; c = -182;
Δ = b2-4ac
Δ = 12-4·1·(-182)
Δ = 729
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{729}=27$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1)-27}{2*1}=\frac{-28}{2}
=-14
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+27}{2*1}=\frac{26}{2}
=13
$