1/3n+3=n-3

Solution for 1/3n+3=n-3 equation:

1/3n+3=n-3

We move all terms to the left:

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


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

-3n^2+18n+1=0

a = -3; b = 18; c = +1;
Δ = b2-4ac
Δ = 182-4·(-3)·1
Δ = 336
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:

$\sqrt{\Delta}=\sqrt{336}=\sqrt{16*21}=\sqrt{16}*\sqrt{21}=4\sqrt{21}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-4\sqrt{21}}{2*-3}=\frac{-18-4\sqrt{21}}{-6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+4\sqrt{21}}{2*-3}=\frac{-18+4\sqrt{21}}{-6}
$