To solve the quadratic equation \(x^2 - 3x + 8 = 0\), we can use the quadratic formula:

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

For the equation \(x^2 - 3x + 8 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -3\)
- \(c = 8\)

Let's substitute these values into the quadratic formula:

\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} \]

This simplifies to:

\[ x = \frac{3 \pm \sqrt{9 - 32}}{2} \]
\[ x = \frac{3 \pm \sqrt{-23}}{2} \]

Since the discriminant (\(\Delta = b^2 - 4ac = -23\)) is negative, the equation has complex roots. We can express the roots as:

\[ x = \frac{3 \pm \sqrt{23}i}{2} \]

Thus, the solutions are:

\[ x = \frac{3 + \sqrt{23}i}{2} \]
\[ x = \frac{3 - \sqrt{23}i}{2} \]

These are the complex roots of the quadratic equation \(x^2 - 3x + 8 = 0\).

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. How do you determine the discriminant of a quadratic equation?
2. What does it mean when the discriminant is negative, zero, or positive?
3. How do you solve quadratic equations by completing the square?
4. What is the importance of complex numbers in quadratic equations?
5. How do you graph the quadratic function \(y = x^2 - 3x + 8\)?
6. Can quadratic equations have real roots and complex roots at the same time?
7. How do you find the vertex of the parabola given by \(y = x^2 - 3x + 8\)?
8. What are some real-life applications of solving quadratic equations?

**Tip:** Always check the discriminant first to determine the nature of the roots before solving a quadratic equation.

Learn how to solve the quadratic equation x^2 - 3x + 8 = 0 with complex roots using the quadratic formula. Understand the importance of complex numbers in quadratic equations. Suitable for students in Grades 10-12.

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