Find the sum of the roots of $\tan^2x-9\tan x+1=0$ that are between $x=0$ and $x=2\pi$ radians.

$ \textbf{(A)}\ \frac{\pi}{2} \qquad\textbf{(B)}\ \pi \qquad\textbf{(C)}\ \frac{3\pi}{2} \qquad\textbf{(D)}\ 3\pi \qquad\textbf{(E)}\ 4\pi $
1. The given equation is a quadratic in terms of $\tan x$:
   \[
   \tan^2 x - 9 \tan x + 1 = 0
   \]
   We can solve this quadratic equation using the quadratic formula:
   \[
   \tan x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \(a = 1\), \(b = -9\), and \(c = 1\). Substituting these values into the quadratic formula, we get:
   \[
   \tan x = \frac{9 \pm \sqrt{81 - 4}}{2} = \frac{9 \pm \sqrt{77}}{2}
   \]
   Therefore, the solutions for \(\tan x\) are:
   \[
   \tan x = \frac{9 + \sqrt{77}}{2} \quad \text{and} \quad \tan x = \frac{9 - \sqrt{77}}{2}
   \]

2. We need to find the values of \(x\) in the interval \([0, 2\pi]\) that satisfy these tangent values. Let:
   \[
   \alpha = \arctan \left( \frac{9 + \sqrt{77}}{2} \right) \quad \text{and} \quad \beta = \arctan \left( \frac{9 - \sqrt{77}}{2} \right)
   \]
   The general solutions for \(x\) are:
   \[
   x = \alpha + k\pi \quad \text{and} \quad x = \beta + k\pi \quad \text{for integer } k
   \]

3. We need to find the specific values of \(x\) within the interval \([0, 2\pi]\):
   \[
   x_1 = \alpha, \quad x_2 = \alpha + \pi, \quad x_3 = \beta, \quad x_4 = \beta + \pi
   \]
   These are the four solutions within the given interval.

4. To find the sum of these roots, we add them together:
   \[
   x_1 + x_2 + x_3 + x_4 = \alpha + (\alpha + \pi) + \beta + (\beta + \pi)
   \]
   Simplifying this, we get:
   \[
   x_1 + x_2 + x_3 + x_4 = 2\alpha + 2\beta + 2\pi
   \]

5. Using the identity for the sum of arctangents, we know:
   \[
   \alpha + \beta = \arctan \left( \frac{\frac{9 + \sqrt{77}}{2} + \frac{9 - \sqrt{77}}{2}}{1 - \left( \frac{9 + \sqrt{77}}{2} \right) \left( \frac{9 - \sqrt{77}}{2} \right)} \right)
   \]
   Simplifying the numerator and denominator:
   \[
   \alpha + \beta = \arctan \left( \frac{9}{1 - \left( \frac{81 - 77}{4} \right)} \right) = \arctan \left( \frac{9}{1 - 1} \right) = \arctan (\infty) = \frac{\pi}{2}
   \]

6. Therefore:
   \[
   2\alpha + 2\beta = 2 \left( \frac{\pi}{2} \right) = \pi
   \]

7. Finally, the sum of the roots is:
   \[
   x_1 + x_2 + x_3 + x_4 = \pi + 2\pi = 3\pi
   \]

The final answer is \(\boxed{3\pi}\)