6/3n=n+5n

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

6/3n=n+5n

We move all terms to the left:

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


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

6/3n-6n=0

We multiply all the terms by the denominator


-6n*3n+6=0

Wy multiply elements

-18n^2+6=0

a = -18; b = 0; c = +6;
Δ = b2-4ac
Δ = 02-4·(-18)·6
Δ = 432
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:

$\sqrt{\Delta}=\sqrt{432}=\sqrt{144*3}=\sqrt{144}*\sqrt{3}=12\sqrt{3}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-12\sqrt{3}}{2*-18}=\frac{0-12\sqrt{3}}{-36}
=-\frac{12\sqrt{3}}{-36}
=-\frac{\sqrt{3}}{-3}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+12\sqrt{3}}{2*-18}=\frac{0+12\sqrt{3}}{-36}
=\frac{12\sqrt{3}}{-36}
=\frac{\sqrt{3}}{-3}
$