1/2n=4n+3

Solution for 1/2n=4n+3 equation:

1/2n=4n+3

We move all terms to the left:

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


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

1/2n-4n-3=0

We multiply all the terms by the denominator


-4n*2n-3*2n+1=0

Wy multiply elements

-8n^2-6n+1=0

a = -8; b = -6; c = +1;
Δ = b2-4ac
Δ = -62-4·(-8)·1
Δ = 68
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:

$\sqrt{\Delta}=\sqrt{68}=\sqrt{4*17}=\sqrt{4}*\sqrt{17}=2\sqrt{17}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{17}}{2*-8}=\frac{6-2\sqrt{17}}{-16}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{17}}{2*-8}=\frac{6+2\sqrt{17}}{-16}
$