2n+6=1/3n

Solution for 2n+6=1/3n equation:

2n+6=1/3n

We move all terms to the left:

2n+6-(1/3n)=0


Domain of the equation: 3n)!=0

n!=0/1

n!=0

n∈R


We add all the numbers together, and all the variables

2n-(+1/3n)+6=0

We get rid of parentheses

2n-1/3n+6=0

We multiply all the terms by the denominator


2n*3n+6*3n-1=0

Wy multiply elements

6n^2+18n-1=0

a = 6; b = 18; c = -1;
Δ = b2-4ac
Δ = 182-4·6·(-1)
Δ = 348
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:

$\sqrt{\Delta}=\sqrt{348}=\sqrt{4*87}=\sqrt{4}*\sqrt{87}=2\sqrt{87}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-2\sqrt{87}}{2*6}=\frac{-18-2\sqrt{87}}{12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+2\sqrt{87}}{2*6}=\frac{-18+2\sqrt{87}}{12}
$