3/5n+1/2=21/3n-2

Solution for 3/5n+1/2=21/3n-2 equation:

3/5n+1/2=21/3n-2

We move all terms to the left:

3/5n+1/2-(21/3n-2)=0


Domain of the equation: 5n!=0

n!=0/5

n!=0

n∈R



Domain of the equation: 3n-2)!=0

n∈R


We get rid of parentheses

3/5n-21/3n+2+1/2=0

We calculate fractions


45n^2/60n^2+36n/60n^2+(-420n)/60n^2+2=0

We multiply all the terms by the denominator


45n^2+36n+(-420n)+2*60n^2=0

Wy multiply elements

45n^2+120n^2+36n+(-420n)=0

We get rid of parentheses

45n^2+120n^2+36n-420n=0

We add all the numbers together, and all the variables

165n^2-384n=0

a = 165; b = -384; c = 0;
Δ = b2-4ac
Δ = -3842-4·165·0
Δ = 147456
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{147456}=384$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-384)-384}{2*165}=\frac{0}{330}
=0
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-384)+384}{2*165}=\frac{768}{330}
=2+18/55
$