To solve the quadratic equation \( x^2 - 2x + 5 = 0 \), we'll use the quadratic formula:

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

For the given equation:

- \( a = 1 \)
- \( b = -2 \)
- \( c = 5 \)

Now, substitute these values into the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(5)}}{2(1)}
\]

Simplify the expression:

\[
x = \frac{2 \pm \sqrt{4 - 20}}{2}
\]

\[
x = \frac{2 \pm \sqrt{-16}}{2}
\]

Since the discriminant \(\sqrt{-16}\) is negative, the roots are complex:

\[
x = \frac{2 \pm 4i}{2}
\]

Finally, simplify the expression:

\[
x = 1 \pm 2i
\]

So the solutions are:

\[
x = 1 + 2i \quad \text{and} \quad x = 1 - 2i
\]

Would you like more details on any part of this solution?

### Follow-up Questions:
1. How do we interpret complex roots in real-world applications?
2. What is the discriminant, and how does it determine the nature of the roots?
3. Can you show how to solve a quadratic equation by completing the square?
4. What happens when the discriminant is zero? What type of roots do we get?
5. How can you graphically represent the equation \(x^2 - 2x + 5 = 0\)?

**Tip:** When the discriminant is negative, it indicates the presence of complex roots.

Learn how to solve the quadratic equation x^2 - 2x + 5 = 0 with complex roots using the quadratic formula. This problem covers key algebraic concepts, including complex numbers and the quadratic formula. Suitable for advanced high school students (Grades 11-12).

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