2+3n=4/5n+12

Solution for 2+3n=4/5n+12 equation:

2+3n=4/5n+12

We move all terms to the left:

2+3n-(4/5n+12)=0


Domain of the equation: 5n+12)!=0

n∈R


We get rid of parentheses

3n-4/5n-12+2=0

We multiply all the terms by the denominator


3n*5n-12*5n+2*5n-4=0

Wy multiply elements

15n^2-60n+10n-4=0

We add all the numbers together, and all the variables

15n^2-50n-4=0

a = 15; b = -50; c = -4;
Δ = b2-4ac
Δ = -502-4·15·(-4)
Δ = 2740
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:

$\sqrt{\Delta}=\sqrt{2740}=\sqrt{4*685}=\sqrt{4}*\sqrt{685}=2\sqrt{685}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-50)-2\sqrt{685}}{2*15}=\frac{50-2\sqrt{685}}{30}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-50)+2\sqrt{685}}{2*15}=\frac{50+2\sqrt{685}}{30}
$