2+1/2n=3n+16

Solution for 2+1/2n=3n+16 equation:

2+1/2n=3n+16

We move all terms to the left:

2+1/2n-(3n+16)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

1/2n-3n-16+2=0

We multiply all the terms by the denominator


-3n*2n-16*2n+2*2n+1=0

Wy multiply elements

-6n^2-32n+4n+1=0

We add all the numbers together, and all the variables

-6n^2-28n+1=0

a = -6; b = -28; c = +1;
Δ = b2-4ac
Δ = -282-4·(-6)·1
Δ = 808
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:

$\sqrt{\Delta}=\sqrt{808}=\sqrt{4*202}=\sqrt{4}*\sqrt{202}=2\sqrt{202}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-28)-2\sqrt{202}}{2*-6}=\frac{28-2\sqrt{202}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-28)+2\sqrt{202}}{2*-6}=\frac{28+2\sqrt{202}}{-12}
$