2/3n=4n-10/3n-12

Solution for 2/3n=4n-10/3n-12 equation:

2/3n=4n-10/3n-12

We move all terms to the left:

2/3n-(4n-10/3n-12)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R



Domain of the equation: 3n-12)!=0

n∈R


We get rid of parentheses

2/3n-4n+10/3n+12=0

We multiply all the terms by the denominator


-4n*3n+12*3n+2+10=0

We add all the numbers together, and all the variables

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

Wy multiply elements

-12n^2+36n+12=0

a = -12; b = 36; c = +12;
Δ = b2-4ac
Δ = 362-4·(-12)·12
Δ = 1872
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:

$\sqrt{\Delta}=\sqrt{1872}=\sqrt{144*13}=\sqrt{144}*\sqrt{13}=12\sqrt{13}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(36)-12\sqrt{13}}{2*-12}=\frac{-36-12\sqrt{13}}{-24}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(36)+12\sqrt{13}}{2*-12}=\frac{-36+12\sqrt{13}}{-24}
$