1/3n=22-n

Solution for 1/3n=22-n equation:

1/3n=22-n

We move all terms to the left:

1/3n-(22-n)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

1/3n-(-1n+22)=0

We get rid of parentheses

1/3n+1n-22=0

We multiply all the terms by the denominator


1n*3n-22*3n+1=0

Wy multiply elements

3n^2-66n+1=0

a = 3; b = -66; c = +1;
Δ = b2-4ac
Δ = -662-4·3·1
Δ = 4344
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:

$\sqrt{\Delta}=\sqrt{4344}=\sqrt{4*1086}=\sqrt{4}*\sqrt{1086}=2\sqrt{1086}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-66)-2\sqrt{1086}}{2*3}=\frac{66-2\sqrt{1086}}{6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-66)+2\sqrt{1086}}{2*3}=\frac{66+2\sqrt{1086}}{6}
$