2/3n+9=n-6

Solution for 2/3n+9=n-6 equation:

2/3n+9=n-6

We move all terms to the left:

2/3n+9-(n-6)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We get rid of parentheses

2/3n-n+6+9=0

We multiply all the terms by the denominator


-n*3n+6*3n+9*3n+2=0

Wy multiply elements

-3n^2+18n+27n+2=0

We add all the numbers together, and all the variables

-3n^2+45n+2=0

a = -3; b = 45; c = +2;
Δ = b2-4ac
Δ = 452-4·(-3)·2
Δ = 2049
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(45)-\sqrt{2049}}{2*-3}=\frac{-45-\sqrt{2049}}{-6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(45)+\sqrt{2049}}{2*-3}=\frac{-45+\sqrt{2049}}{-6}
$