3n-5=9+1/2n

Solution for 3n-5=9+1/2n equation:

3n-5=9+1/2n

We move all terms to the left:

3n-5-(9+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

3n-(1/2n+9)-5=0

We get rid of parentheses

3n-1/2n-9-5=0

We multiply all the terms by the denominator


3n*2n-9*2n-5*2n-1=0

Wy multiply elements

6n^2-18n-10n-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}
$