3/3+3n=4/5n+18

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

3/3+3n=4/5n+18

We move all terms to the left:

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


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

n∈R


We add all the numbers together, and all the variables

3n-(4/5n+18)+1=0

We get rid of parentheses

3n-4/5n-18+1=0

We multiply all the terms by the denominator


3n*5n-18*5n+1*5n-4=0

Wy multiply elements

15n^2-90n+5n-4=0

We add all the numbers together, and all the variables

15n^2-85n-4=0

a = 15; b = -85; c = -4;
Δ = b2-4ac
Δ = -852-4·15·(-4)
Δ = 7465
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-85)-\sqrt{7465}}{2*15}=\frac{85-\sqrt{7465}}{30}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-85)+\sqrt{7465}}{2*15}=\frac{85+\sqrt{7465}}{30}
$