222/3+3n=4/5n+12

Solution for 222/3+3n=4/5n+12 equation:

222/3+3n=4/5n+12

We move all terms to the left:

222/3+3n-(4/5n+12)=0


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

n∈R


We add all the numbers together, and all the variables

3n-(4/5n+12)+74=0

We get rid of parentheses

3n-4/5n-12+74=0

We multiply all the terms by the denominator


3n*5n-12*5n+74*5n-4=0

Wy multiply elements

15n^2-60n+370n-4=0

We add all the numbers together, and all the variables

15n^2+310n-4=0

a = 15; b = 310; c = -4;
Δ = b2-4ac
Δ = 3102-4·15·(-4)
Δ = 96340
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:

$\sqrt{\Delta}=\sqrt{96340}=\sqrt{4*24085}=\sqrt{4}*\sqrt{24085}=2\sqrt{24085}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(310)-2\sqrt{24085}}{2*15}=\frac{-310-2\sqrt{24085}}{30}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(310)+2\sqrt{24085}}{2*15}=\frac{-310+2\sqrt{24085}}{30}
$