2-1/3n=3n+16

Solution for 2-1/3n=3n+16 equation:

2-1/3n=3n+16

We move all terms to the left:

2-1/3n-(3n+16)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We get rid of parentheses

-1/3n-3n-16+2=0

We multiply all the terms by the denominator


-3n*3n-16*3n+2*3n-1=0

Wy multiply elements

-9n^2-48n+6n-1=0

We add all the numbers together, and all the variables

-9n^2-42n-1=0

a = -9; b = -42; c = -1;
Δ = b2-4ac
Δ = -422-4·(-9)·(-1)
Δ = 1728
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:

$\sqrt{\Delta}=\sqrt{1728}=\sqrt{576*3}=\sqrt{576}*\sqrt{3}=24\sqrt{3}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-24\sqrt{3}}{2*-9}=\frac{42-24\sqrt{3}}{-18}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+24\sqrt{3}}{2*-9}=\frac{42+24\sqrt{3}}{-18}
$