2/3n+9=n+1

Solution for 2/3n+9=n+1 equation:

2/3n+9=n+1

We move all terms to the left:

2/3n+9-(n+1)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We get rid of parentheses

2/3n-n-1+9=0

We multiply all the terms by the denominator


-n*3n-1*3n+9*3n+2=0

Wy multiply elements

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

We add all the numbers together, and all the variables

-3n^2+24n+2=0

a = -3; b = 24; c = +2;
Δ = b2-4ac
Δ = 242-4·(-3)·2
Δ = 600
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:

$\sqrt{\Delta}=\sqrt{600}=\sqrt{100*6}=\sqrt{100}*\sqrt{6}=10\sqrt{6}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(24)-10\sqrt{6}}{2*-3}=\frac{-24-10\sqrt{6}}{-6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(24)+10\sqrt{6}}{2*-3}=\frac{-24+10\sqrt{6}}{-6}
$