2/9+1/3n=3/2n+1

Solution for 2/9+1/3n=3/2n+1 equation:

2/9+1/3n=3/2n+1

We move all terms to the left:

2/9+1/3n-(3/2n+1)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R



Domain of the equation: 2n+1)!=0

n∈R


We get rid of parentheses

1/3n-3/2n-1+2/9=0

We calculate fractions


24n^2/486n^2+162n/486n^2+(-729n)/486n^2-1=0

We multiply all the terms by the denominator


24n^2+162n+(-729n)-1*486n^2=0

Wy multiply elements

24n^2-486n^2+162n+(-729n)=0

We get rid of parentheses

24n^2-486n^2+162n-729n=0

We add all the numbers together, and all the variables

-462n^2-567n=0

a = -462; b = -567; c = 0;
Δ = b2-4ac
Δ = -5672-4·(-462)·0
Δ = 321489
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{321489}=567$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-567)-567}{2*-462}=\frac{0}{-924}
=0
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-567)+567}{2*-462}=\frac{1134}{-924}
=-1+5/22
$