16+1/3n=3n

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

16+1/3n=3n

We move all terms to the left:

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


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

a = -9; b = 48; c = +1;
Δ = b2-4ac
Δ = 482-4·(-9)·1
Δ = 2340
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:

$\sqrt{\Delta}=\sqrt{2340}=\sqrt{36*65}=\sqrt{36}*\sqrt{65}=6\sqrt{65}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(48)-6\sqrt{65}}{2*-9}=\frac{-48-6\sqrt{65}}{-18}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(48)+6\sqrt{65}}{2*-9}=\frac{-48+6\sqrt{65}}{-18}
$