3n(n+1)=27

Solution for 3n(n+1)=27 equation:

3n(n+1)=27

We move all terms to the left:

3n(n+1)-(27)=0

We multiply parentheses

3n^2+3n-27=0

a = 3; b = 3; c = -27;
Δ = b2-4ac
Δ = 32-4·3·(-27)
Δ = 333
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:

$\sqrt{\Delta}=\sqrt{333}=\sqrt{9*37}=\sqrt{9}*\sqrt{37}=3\sqrt{37}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(3)-3\sqrt{37}}{2*3}=\frac{-3-3\sqrt{37}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(3)+3\sqrt{37}}{2*3}=\frac{-3+3\sqrt{37}}{6}
$