To complete the square for the quadratic equation \(x^2 + 6x + 11 = 0\), follow these steps:

1. **Start with the equation**:
   \[
   x^2 + 6x + 11 = 0
   \]

2. **Move the constant to the other side**:
   \[
   x^2 + 6x = -11
   \]

3. **Complete the square** by adding \(\left(\frac{6}{2}\right)^2 = 9\) to both sides:
   \[
   x^2 + 6x + 9 = -11 + 9
   \]
   \[
   (x + 3)^2 = -2
   \]

4. **Take the square root of both sides**:
   \[
   x + 3 = \pm \sqrt{-2}
   \]
   \[
   x + 3 = \pm i\sqrt{2}
   \]

5. **Solve for \(x\)**:
   \[
   x = -3 \pm i\sqrt{2}
   \]

So, the solutions are:
\[
x = -3 + i\sqrt{2} \quad \text{or} \quad x = -3 - i\sqrt{2}
\]

Would you like more details on any step, or do you have questions?

Here are five related questions for further exploration:
1. How do you complete the square for any quadratic equation?
2. What is the geometric interpretation of completing the square?
3. How can you solve a quadratic equation using the quadratic formula?
4. What are complex numbers, and how do they arise in equations like this?
5. How do real and complex solutions of quadratics differ?

**Tip**: Completing the square is not only useful for solving quadratics but also for understanding the vertex form of a parabola.

Master the technique of completing the square for the quadratic equation x² + 6x + 11 = 0 with a comprehensive step-by-step solution. Explore key algebraic concepts, including complex numbers and the Fundamental Theorem of Algebra. Suitable for students in Grades 9-12.

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