2/3n-111,n=12

Solution for 2/3n-111,n=12 equation:

2/3n-111.n=12

We move all terms to the left:

2/3n-111.n-(12)=0


Domain of the equation: 3n!=0

n!=0/3

n!=0

n∈R


We add all the numbers together, and all the variables

-111n+2/3n-12=0

We multiply all the terms by the denominator


-111n*3n-12*3n+2=0

Wy multiply elements

-333n^2-36n+2=0

a = -333; b = -36; c = +2;
Δ = b2-4ac
Δ = -362-4·(-333)·2
Δ = 3960
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:

$\sqrt{\Delta}=\sqrt{3960}=\sqrt{36*110}=\sqrt{36}*\sqrt{110}=6\sqrt{110}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-36)-6\sqrt{110}}{2*-333}=\frac{36-6\sqrt{110}}{-666}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-36)+6\sqrt{110}}{2*-333}=\frac{36+6\sqrt{110}}{-666}
$