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

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

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

Wy multiply elements

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

a = -6; b = -32; c = +2;
Δ = b2-4ac
Δ = -322-4·(-6)·2
Δ = 1072
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:

$\sqrt{\Delta}=\sqrt{1072}=\sqrt{16*67}=\sqrt{16}*\sqrt{67}=4\sqrt{67}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-32)-4\sqrt{67}}{2*-6}=\frac{32-4\sqrt{67}}{-12}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-32)+4\sqrt{67}}{2*-6}=\frac{32+4\sqrt{67}}{-12}
$