To solve the quadratic equation \(x^2 + 2x + 6 = 0\), we can use the quadratic formula:

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

Here, the coefficients are:
- \( a = 1 \)
- \( b = 2 \)
- \( c = 6 \)

First, we calculate the discriminant \( \Delta \):

\[ \Delta = b^2 - 4ac \]
\[ \Delta = 2^2 - 4 \cdot 1 \cdot 6 \]
\[ \Delta = 4 - 24 \]
\[ \Delta = -20 \]

Since the discriminant is negative, the quadratic equation has no real roots but has two complex roots.

Next, we apply the quadratic formula:

\[ x = \frac{-2 \pm \sqrt{-20}}{2 \cdot 1} \]
\[ x = \frac{-2 \pm \sqrt{-20}}{2} \]
\[ x = \frac{-2 \pm \sqrt{20i}}{2} \]
\[ x = \frac{-2 \pm 2\sqrt{5}i}{2} \]
\[ x = -1 \pm \sqrt{5}i \]

Therefore, the solutions to the quadratic equation \(x^2 + 2x + 6 = 0\) are:

\[ x = -1 + \sqrt{5}i \]
\[ x = -1 - \sqrt{5}i \]

Would you like more details or have any questions?

Here are 8 related questions:
1. What are the steps to derive the quadratic formula?
2. How do you determine if a quadratic equation has real or complex roots?
3. What is the significance of the discriminant in a quadratic equation?
4. How can you solve a quadratic equation using the completing the square method?
5. What are the different forms of a quadratic equation?
6. Can you provide an example of a quadratic equation with real roots?
7. How do you factorize a quadratic equation?
8. What is the vertex form of a quadratic equation?

**Tip:** Always check the discriminant first when solving quadratic equations to determine the nature of the roots (real or complex).

Learn how to solve the quadratic equation x^2 + 2x + 6 = 0 with complex roots using the quadratic formula. Understand the discriminant and its role in determining the nature of roots. Suitable for advanced high school students.

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