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The problem asks us to find the solution set of the quadratic equation \(3x^2 - 12x - 1 = 0\) by completing the square. Let's solve it step by step.

### Step-by-Step Solution:

1. **Write the equation in the form \(ax^2 + bx + c = 0\):**

   The given quadratic equation is:
   \[
   3x^2 - 12x - 1 = 0
   \]

2. **Divide all terms by the coefficient of \(x^2\) (which is 3) to simplify:**

   \[
   x^2 - 4x - \frac{1}{3} = 0
   \]

3. **Move the constant term to the right side:**

   \[
   x^2 - 4x = \frac{1}{3}
   \]

4. **To complete the square, add \(\left(\frac{b}{2}\right)^2\) to both sides, where \(b = -4\):**

   First, calculate \(\left(\frac{b}{2}\right)^2\):
   \[
   \left(\frac{-4}{2}\right)^2 = (-2)^2 = 4
   \]

   Add 4 to both sides:
   \[
   x^2 - 4x + 4 = \frac{1}{3} + 4
   \]

5. **Simplify the equation:**

   \[
   x^2 - 4x + 4 = \frac{1}{3} + \frac{12}{3} = \frac{13}{3}
   \]

   Now, the left side can be factored as a perfect square:
   \[
   (x - 2)^2 = \frac{13}{3}
   \]

6. **Take the square root of both sides:**

   \[
   x - 2 = \pm \sqrt{\frac{13}{3}}
   \]

   Simplify the square root:
   \[
   x - 2 = \pm \frac{\sqrt{13}}{\sqrt{3}} = \pm \frac{\sqrt{39}}{3}
   \]

7. **Solve for \(x\):**

   \[
   x = 2 \pm \frac{\sqrt{39}}{3}
   \]

### Solution Set:

The solutions for the quadratic equation \(3x^2 - 12x - 1 = 0\) are:
\[
x = 2 + \frac{\sqrt{39}}{3} \quad \text{and} \quad x = 2 - \frac{\sqrt{39}}{3}
\]

Would you like more details or have any questions about the solution? Here are some additional topics that could expand your understanding:

1. How does the method of completing the square compare to the quadratic formula?
2. What is the geometric interpretation of completing the square?
3. Why is it important to know different methods for solving quadratic equations?
4. Can completing the square be used to derive the quadratic formula?
5. How would you solve the quadratic equation if the coefficient of \(x^2\) was negative?

**Tip:** Completing the square is a powerful technique not just for solving quadratics, but also for understanding the nature of the parabola represented by the equation.

The query submitted by the user: How to solve the quadratic equation 3x^2 - 12x - 1 = 0 by completing the square?

Learn how to solve the quadratic equation 3x^2 - 12x - 1 = 0 using the method of completing the square with a step-by-step explanation. This problem involves algebraic concepts such as completing the square, square roots, and the quadratic equation formula. Suitable for high school students in Grades 10-12.

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