3*4n=8-1/2n

Solution for 3*4n=8-1/2n equation:

3*4n=8-1/2n

We move all terms to the left:

3*4n-(8-1/2n)=0


Domain of the equation: 2n)!=0

n!=0/1

n!=0

n∈R


We add all the numbers together, and all the variables

3*4n-(-1/2n+8)=0

Wy multiply elements

12n-(-1/2n+8)=0

We get rid of parentheses

12n+1/2n-8=0

We multiply all the terms by the denominator


12n*2n-8*2n+1=0

Wy multiply elements

24n^2-16n+1=0

a = 24; b = -16; c = +1;
Δ = b2-4ac
Δ = -162-4·24·1
Δ = 160
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:

$\sqrt{\Delta}=\sqrt{160}=\sqrt{16*10}=\sqrt{16}*\sqrt{10}=4\sqrt{10}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-16)-4\sqrt{10}}{2*24}=\frac{16-4\sqrt{10}}{48}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-16)+4\sqrt{10}}{2*24}=\frac{16+4\sqrt{10}}{48}
$