-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)+1854n+(-2232n)=0

We get rid of parentheses

-1836n^2+1854n-2232n=0

We add all the numbers together, and all the variables

-1836n^2-378n=0

a = -1836; b = -378; c = 0;
Δ = b2-4ac
Δ = -3782-4·(-1836)·0
Δ = 142884
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{142884}=378$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-378)-378}{2*-1836}=\frac{0}{-3672}
=0
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-378)+378}{2*-1836}=\frac{756}{-3672}
=-7/34
$