We are given the equation $y = \frac{2(\sqrt{x^2}+m)}{3N}$ and we want to express $x$ in terms of $y$, $N$, and $m$.

1. Problem Description

We are given the equation $y = \frac{2(\sqrt{x^2}+m)}{3N}$ and we want to express $x$ in terms of $y$, $N$, and $m$.

2. Solution Steps

First, we isolate the term containing the square root:

$y = \frac{2(\sqrt{x^2+m})}{3N}$

$3Ny = 2\sqrt{x^2+m}$

Divide both sides by 2:

$\frac{3Ny}{2} = \sqrt{x^2+m}$

Square both sides:

$(\frac{3Ny}{2})^2 = (\sqrt{x^2+m})^2$

$\frac{9N^2y^2}{4} = x^2+m$

Isolate $x^2$:

$x^2 = \frac{9N^2y^2}{4} - m$

$x^2 = \frac{9N^2y^2 - 4m}{4}$

Take the square root of both sides:

$x = \sqrt{\frac{9N^2y^2 - 4m}{4}}$

$x = \frac{\sqrt{9N^2y^2 - 4m}}{\sqrt{4}}$

$x = \frac{\sqrt{9N^2y^2 - 4m}}{2}$

3. Final Answer

The final answer is C. $\frac{\sqrt{9y^2N^2 - 4m}}{2}$