2/3n=4n-6

Solution for 2/3n=4n-6 equation:

2/3n=4n-6

We move all terms to the left:

2/3n-(4n-6)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We get rid of parentheses

2/3n-4n+6=0

We multiply all the terms by the denominator


-4n*3n+6*3n+2=0

Wy multiply elements

-12n^2+18n+2=0

a = -12; b = 18; c = +2;
Δ = b2-4ac
Δ = 182-4·(-12)·2
Δ = 420
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:

$\sqrt{\Delta}=\sqrt{420}=\sqrt{4*105}=\sqrt{4}*\sqrt{105}=2\sqrt{105}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-2\sqrt{105}}{2*-12}=\frac{-18-2\sqrt{105}}{-24}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+2\sqrt{105}}{2*-12}=\frac{-18+2\sqrt{105}}{-24}
$