2n=3/3n+5

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

2n=3/3n+5

We move all terms to the left:

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


Domain of the equation: 3n+5)!=0

n∈R


We get rid of parentheses

2n-3/3n-5=0

We multiply all the terms by the denominator


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

Wy multiply elements

6n^2-15n-3=0

a = 6; b = -15; c = -3;
Δ = b2-4ac
Δ = -152-4·6·(-3)
Δ = 297
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{297}=\sqrt{9*33}=\sqrt{9}*\sqrt{33}=3\sqrt{33}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-15)-3\sqrt{33}}{2*6}=\frac{15-3\sqrt{33}}{12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-15)+3\sqrt{33}}{2*6}=\frac{15+3\sqrt{33}}{12}
$