1/2n+1=n+3

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

1/2n+1=n+3

We move all terms to the left:

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


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

-2n^2-4n+1=0

a = -2; b = -4; c = +1;
Δ = b2-4ac
Δ = -42-4·(-2)·1
Δ = 24
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:

$\sqrt{\Delta}=\sqrt{24}=\sqrt{4*6}=\sqrt{4}*\sqrt{6}=2\sqrt{6}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{6}}{2*-2}=\frac{4-2\sqrt{6}}{-4}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{6}}{2*-2}=\frac{4+2\sqrt{6}}{-4}
$