1+3n=4/5n+12

Solution for 1+3n=4/5n+12 equation:

1+3n=4/5n+12

We move all terms to the left:

1+3n-(4/5n+12)=0


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

n∈R


We get rid of parentheses

3n-4/5n-12+1=0

We multiply all the terms by the denominator


3n*5n-12*5n+1*5n-4=0

Wy multiply elements

15n^2-60n+5n-4=0

We add all the numbers together, and all the variables

15n^2-55n-4=0

a = 15; b = -55; c = -4;
Δ = b2-4ac
Δ = -552-4·15·(-4)
Δ = 3265
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{-(-55)-\sqrt{3265}}{2*15}=\frac{55-\sqrt{3265}}{30}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-55)+\sqrt{3265}}{2*15}=\frac{55+\sqrt{3265}}{30}
$