Let `k` be sum of all x in the interval `[0, 2pi]` such that `3 cot^(2) x+8 cot x+3=0`, then the value of `k//pi` is ___________.

Explanation

To solve the equation \( 3 \cot^2 x + 8 \cot x + 3 = 0 \) and find the sum of all \( x \) in the interval \([0, 2\pi]\) such that this equation holds, we can follow these steps:

### Step 1: Substitute \( y = \cot x \)
We start by letting \( y = \cot x \). This transforms our equation into:
\[ 3y^2 + 8y + 3 = 0 \]

### Step 2: Use the quadratic formula
To find the roots of the quadratic equation, we apply the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 3, b = 8, c = 3 \).

Calculating the discriminant:
\[ b^2 - 4ac = 8^2 - 4 \cdot 3 \cdot 3 = 64 - 36 = 28 \]

Now substituting back into the formula:
\[ y = \frac{-8 \pm \sqrt{28}}{2 \cdot 3} = \frac{-8 \pm 2\sqrt{7}}{6} = \frac{-4 \pm \sqrt{7}}{3} \]

### Step 3: Find the values of \( y \)
Thus, the roots are:
\[ y_1 = \frac{-4 + \sqrt{7}}{3}, \quad y_2 = \frac{-4 - \sqrt{7}}{3} \]

### Step 4: Find the corresponding \( x \) values
Since \( y = \cot x \), we need to find \( x \) such that:
\[ \cot x = y_1 \quad \text{and} \quad \cot x = y_2 \]

The general solutions for \( x \) in terms of \( y \) are:
\[ x = \cot^{-1}(y) + n\pi \quad \text{for } n \in \mathbb{Z} \]

### Step 5: Calculate \( x \) values in the interval \([0, 2\pi]\)
For \( y_1 \):
1. \( x_1 = \cot^{-1}\left(\frac{-4 + \sqrt{7}}{3}\right) \)
2. \( x_2 = \cot^{-1}\left(\frac{-4 + \sqrt{7}}{3}\right) + \pi \)

For \( y_2 \):
1. \( x_3 = \cot^{-1}\left(\frac{-4 - \sqrt{7}}{3}\right) \)
2. \( x_4 = \cot^{-1}\left(\frac{-4 - \sqrt{7}}{3}\right) + \pi \)

### Step 6: Sum the \( x \) values
The sum of all \( x \) values is:
\[ k = x_1 + x_2 + x_3 + x_4 \]
\[ = \cot^{-1}\left(\frac{-4 + \sqrt{7}}{3}\right) + \left(\cot^{-1}\left(\frac{-4 + \sqrt{7}}{3}\right) + \pi\right) + \cot^{-1}\left(\frac{-4 - \sqrt{7}}{3}\right) + \left(\cot^{-1}\left(\frac{-4 - \sqrt{7}}{3}\right) + \pi\right) \]
\[ = 2\cot^{-1}\left(\frac{-4 + \sqrt{7}}{3}\right) + 2\cot^{-1}\left(\frac{-4 - \sqrt{7}}{3}\right) + 2\pi \]
\[ = 2\left(\cot^{-1}\left(\frac{-4 + \sqrt{7}}{3}\right) + \cot^{-1}\left(\frac{-4 - \sqrt{7}}{3}\right)\right) + 2\pi \]

Using the property \( \cot^{-1}(a) + \cot^{-1}(b) = \pi \) when \( ab < 1 \) (which holds here), we find:
\[ k = 2\pi + 2\pi = 4\pi \]

### Step 7: Find \( \frac{k}{\pi} \)
Finally, we compute:
\[ \frac{k}{\pi} = \frac{4\pi}{\pi} = 4 \]

Thus, the value of \( \frac{k}{\pi} \) is \( \boxed{4} \).