The equation is:

$3x^2 - 6x + 11 = 0$

This is a quadratic equation. To solve it, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 3$, $b = -6$, and $c = 11$.

Step 1: Calculate the discriminant ($\Delta$)

$\Delta = b^2 - 4ac$ $\Delta = (-6)^2 - 4(3)(11)$ $\Delta = 36 - 132 = -96$

Step 2: Analyze the discriminant

Since $\Delta < 0$, the roots are complex numbers.

Step 3: Solve for $x$

The formula for complex roots is: $x = \frac{-b \pm i\sqrt{|\Delta|}}{2a}$ Substitute the values: $x = \frac{-(-6) \pm i\sqrt{96}}{2(3)}$ $x = \frac{6 \pm i\sqrt{96}}{6}$

Simplify further: $\sqrt{96} = \sqrt{16 \cdot 6} = 4\sqrt{6}$ $x = \frac{6 \pm i \cdot 4\sqrt{6}}{6}$ $x = 1 \pm \frac{2i\sqrt{6}}{3}$

Final Answer:

The solutions are: $x = 1 + \frac{2i\sqrt{6}}{3}, \quad x = 1 - \frac{2i\sqrt{6}}{3}$

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Do you want more details or clarification? 😊
Here are 5 related questions for you to explore further:

1. What does the discriminant tell us about the nature of the roots?
2. How can we represent complex roots graphically?
3. Can you solve another quadratic equation with different coefficients?
4. How is the quadratic formula derived from completing the square?
5. What are some real-world applications of quadratic equations?

Tip: Always check the discriminant first to determine the nature of the roots before solving!