31/3n+1/3+11/2n=51/6

Solution for 31/3n+1/3+11/2n=51/6 equation:

31/3n+1/3+11/2n=51/6

We move all terms to the left:

31/3n+1/3+11/2n-(51/6)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R



Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We add all the numbers together, and all the variables

31/3n+11/2n+1/3-(+51/6)=0

We get rid of parentheses

31/3n+11/2n+1/3-51/6=0

We calculate fractions


(-1836n^2)/324n^2+2232n/324n^2+1782n/324n^2+72n/324n^2=0

We multiply all the terms by the denominator


(-1836n^2)+2232n+1782n+72n=0

We add all the numbers together, and all the variables

(-1836n^2)+4086n=0

We get rid of parentheses

-1836n^2+4086n=0

a = -1836; b = 4086; c = 0;
Δ = b2-4ac
Δ = 40862-4·(-1836)·0
Δ = 16695396
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{16695396}=4086$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4086)-4086}{2*-1836}=\frac{-8172}{-3672}
=2+23/102
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4086)+4086}{2*-1836}=\frac{0}{-3672}
=0
$