2/3n=5n+39

Solution for 2/3n=5n+39 equation:

2/3n=5n+39

We move all terms to the left:

2/3n-(5n+39)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We get rid of parentheses

2/3n-5n-39=0

We multiply all the terms by the denominator


-5n*3n-39*3n+2=0

Wy multiply elements

-15n^2-117n+2=0

a = -15; b = -117; c = +2;
Δ = b2-4ac
Δ = -1172-4·(-15)·2
Δ = 13809
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{-(-117)-\sqrt{13809}}{2*-15}=\frac{117-\sqrt{13809}}{-30}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-117)+\sqrt{13809}}{2*-15}=\frac{117+\sqrt{13809}}{-30}
$