12-n=1/3n

Solution for 12-n=1/3n equation:

12-n=1/3n

We move all terms to the left:

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


Domain of the equation: 3n)!=0

n!=0/1

n!=0

n∈R


We add all the numbers together, and all the variables

-n-(+1/3n)+12=0

We add all the numbers together, and all the variables

-1n-(+1/3n)+12=0

We get rid of parentheses

-1n-1/3n+12=0

We multiply all the terms by the denominator


-1n*3n+12*3n-1=0

Wy multiply elements

-3n^2+36n-1=0

a = -3; b = 36; c = -1;
Δ = b2-4ac
Δ = 362-4·(-3)·(-1)
Δ = 1284
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:

$\sqrt{\Delta}=\sqrt{1284}=\sqrt{4*321}=\sqrt{4}*\sqrt{321}=2\sqrt{321}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(36)-2\sqrt{321}}{2*-3}=\frac{-36-2\sqrt{321}}{-6}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(36)+2\sqrt{321}}{2*-3}=\frac{-36+2\sqrt{321}}{-6}
$