3n(3n+4)=54+6n

Solution for 3n(3n+4)=54+6n equation:

3n(3n+4)=54+6n

We move all terms to the left:

3n(3n+4)-(54+6n)=0

We add all the numbers together, and all the variables

3n(3n+4)-(6n+54)=0

We multiply parentheses

9n^2+12n-(6n+54)=0

We get rid of parentheses

9n^2+12n-6n-54=0

We add all the numbers together, and all the variables

9n^2+6n-54=0

a = 9; b = 6; c = -54;
Δ = b2-4ac
Δ = 62-4·9·(-54)
Δ = 1980
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:

$\sqrt{\Delta}=\sqrt{1980}=\sqrt{36*55}=\sqrt{36}*\sqrt{55}=6\sqrt{55}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-6\sqrt{55}}{2*9}=\frac{-6-6\sqrt{55}}{18}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+6\sqrt{55}}{2*9}=\frac{-6+6\sqrt{55}}{18}
$