13/3n=4/5n+18

Solution for 13/3n=4/5n+18 equation:

13/3n=4/5n+18

We move all terms to the left:

13/3n-(4/5n+18)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R



Domain of the equation: 5n+18)!=0

n∈R


We get rid of parentheses

13/3n-4/5n-18=0

We calculate fractions


65n/15n^2+(-12n)/15n^2-18=0

We multiply all the terms by the denominator


65n+(-12n)-18*15n^2=0

Wy multiply elements

-270n^2+65n+(-12n)=0

We get rid of parentheses

-270n^2+65n-12n=0

We add all the numbers together, and all the variables

-270n^2+53n=0

a = -270; b = 53; c = 0;
Δ = b2-4ac
Δ = 532-4·(-270)·0
Δ = 2809
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{2809}=53$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(53)-53}{2*-270}=\frac{-106}{-540}
=53/270
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(53)+53}{2*-270}=\frac{0}{-540}
=0
$