2/3n-6=n

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

2/3n-6=n

We move all terms to the left:

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


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

-1n+2/3n-6=0

We multiply all the terms by the denominator


-1n*3n-6*3n+2=0

Wy multiply elements

-3n^2-18n+2=0

a = -3; b = -18; c = +2;
Δ = b2-4ac
Δ = -182-4·(-3)·2
Δ = 348
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:

$\sqrt{\Delta}=\sqrt{348}=\sqrt{4*87}=\sqrt{4}*\sqrt{87}=2\sqrt{87}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{87}}{2*-3}=\frac{18-2\sqrt{87}}{-6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{87}}{2*-3}=\frac{18+2\sqrt{87}}{-6}
$