2-1/2n=3n+6

Solution for 2-1/2n=3n+6 equation:

2-1/2n=3n+6

We move all terms to the left:

2-1/2n-(3n+6)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

-1/2n-3n-6+2=0

We multiply all the terms by the denominator


-3n*2n-6*2n+2*2n-1=0

Wy multiply elements

-6n^2-12n+4n-1=0

We add all the numbers together, and all the variables

-6n^2-8n-1=0

a = -6; b = -8; c = -1;
Δ = b2-4ac
Δ = -82-4·(-6)·(-1)
Δ = 40
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:

$\sqrt{\Delta}=\sqrt{40}=\sqrt{4*10}=\sqrt{4}*\sqrt{10}=2\sqrt{10}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{10}}{2*-6}=\frac{8-2\sqrt{10}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{10}}{2*-6}=\frac{8+2\sqrt{10}}{-12}
$