3n+5=1/6n=

Solution for 3n+5=1/6n= equation:

3n+5=1/6n=

We move all terms to the left:

3n+5-(1/6n)=0


Domain of the equation: 6n)!=0

n!=0/1

n!=0

n∈R


We add all the numbers together, and all the variables

3n-(+1/6n)+5=0

We get rid of parentheses

3n-1/6n+5=0

We multiply all the terms by the denominator


3n*6n+5*6n-1=0

Wy multiply elements

18n^2+30n-1=0

a = 18; b = 30; c = -1;
Δ = b2-4ac
Δ = 302-4·18·(-1)
Δ = 972
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:

$\sqrt{\Delta}=\sqrt{972}=\sqrt{324*3}=\sqrt{324}*\sqrt{3}=18\sqrt{3}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(30)-18\sqrt{3}}{2*18}=\frac{-30-18\sqrt{3}}{36}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(30)+18\sqrt{3}}{2*18}=\frac{-30+18\sqrt{3}}{36}
$