5n-3=1/3n+2

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

5n-3=1/3n+2

We move all terms to the left:

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


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

n∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


5n*3n-2*3n-3*3n-1=0

Wy multiply elements

15n^2-6n-9n-1=0

We add all the numbers together, and all the variables

15n^2-15n-1=0

a = 15; b = -15; c = -1;
Δ = b2-4ac
Δ = -152-4·15·(-1)
Δ = 285
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{-(-15)-\sqrt{285}}{2*15}=\frac{15-\sqrt{285}}{30}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-15)+\sqrt{285}}{2*15}=\frac{15+\sqrt{285}}{30}
$