n-6/3n+n-5/3n=1

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

n-6/3n+n-5/3n=1

We move all terms to the left:

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


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

2n-6/3n-5/3n-1=0

We multiply all the terms by the denominator


2n*3n-1*3n-6-5=0

We add all the numbers together, and all the variables

2n*3n-1*3n-11=0

Wy multiply elements

6n^2-3n-11=0

a = 6; b = -3; c = -11;
Δ = b2-4ac
Δ = -32-4·6·(-11)
Δ = 273
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-\sqrt{273}}{2*6}=\frac{3-\sqrt{273}}{12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+\sqrt{273}}{2*6}=\frac{3+\sqrt{273}}{12}
$