1/3n=n-22

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

1/3n=n-22

We move all terms to the left:

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


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We get rid of parentheses

1/3n-n+22=0

We multiply all the terms by the denominator


-n*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 = 4368
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:

$\sqrt{\Delta}=\sqrt{4368}=\sqrt{16*273}=\sqrt{16}*\sqrt{273}=4\sqrt{273}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(66)-4\sqrt{273}}{2*-3}=\frac{-66-4\sqrt{273}}{-6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(66)+4\sqrt{273}}{2*-3}=\frac{-66+4\sqrt{273}}{-6}
$