1/2n+3=47n*3

Solution for 1/2n+3=47n*3 equation:

1/2n+3=47n*3

We move all terms to the left:

1/2n+3-(47n*3)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We add all the numbers together, and all the variables

1/2n-(+47n*3)+3=0

We get rid of parentheses

1/2n-47n*3+3=0

We multiply all the terms by the denominator


-(47n*3)*2n+3*2n+1=0

We add all the numbers together, and all the variables

-(+47n*3)*2n+3*2n+1=0

We multiply parentheses

-282n^2+3*2n+1=0

Wy multiply elements

-282n^2+6n+1=0

a = -282; b = 6; c = +1;
Δ = b2-4ac
Δ = 62-4·(-282)·1
Δ = 1164
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:

$\sqrt{\Delta}=\sqrt{1164}=\sqrt{4*291}=\sqrt{4}*\sqrt{291}=2\sqrt{291}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{291}}{2*-282}=\frac{-6-2\sqrt{291}}{-564}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{291}}{2*-282}=\frac{-6+2\sqrt{291}}{-564}
$