-10/3n+1/3+3/2n=-31/6

Solution for -10/3n+1/3+3/2n=-31/6 equation:

-10/3n+1/3+3/2n=-31/6

We move all terms to the left:

-10/3n+1/3+3/2n-(-31/6)=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 get rid of parentheses

-10/3n+3/2n+1/3+31/6=0

We calculate fractions


1116n^2/324n^2+(-720n)/324n^2+486n/324n^2+72n/324n^2=0

We multiply all the terms by the denominator


1116n^2+(-720n)+486n+72n=0

We add all the numbers together, and all the variables

1116n^2+558n+(-720n)=0

We get rid of parentheses

1116n^2+558n-720n=0

We add all the numbers together, and all the variables

1116n^2-162n=0

a = 1116; b = -162; c = 0;
Δ = b2-4ac
Δ = -1622-4·1116·0
Δ = 26244
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{26244}=162$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-162)-162}{2*1116}=\frac{0}{2232}
=0
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-162)+162}{2*1116}=\frac{324}{2232}
=9/62
$