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

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

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

We move all terms to the left:

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

We add all the numbers together, and all the variables

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

We multiply parentheses

3n^2-2n/3-6=0

We multiply all the terms by the denominator


3n^2*3-2n-6*3=0

We add all the numbers together, and all the variables

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

Wy multiply elements

9n^2-2n-18=0

a = 9; b = -2; c = -18;
Δ = b2-4ac
Δ = -22-4·9·(-18)
Δ = 652
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:

$\sqrt{\Delta}=\sqrt{652}=\sqrt{4*163}=\sqrt{4}*\sqrt{163}=2\sqrt{163}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{163}}{2*9}=\frac{2-2\sqrt{163}}{18}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{163}}{2*9}=\frac{2+2\sqrt{163}}{18}
$