To solve the quadratic equation \(x^2 - 3x + 8 = 0\) by completing the square, follow these steps:

1. **Rewrite the equation:** Move the constant term to the other side of the equation.
   \[
   x^2 - 3x = -8
   \]

2. **Complete the square:** To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides of the equation. The coefficient of \(x\) is \(-3\), so half of it is \(-\frac{3}{2}\), and squaring it gives \(\left(-\frac{3}{2}\right)^2 = \frac{9}{4}\).

   Add \(\frac{9}{4}\) to both sides:
   \[
   x^2 - 3x + \frac{9}{4} = -8 + \frac{9}{4}
   \]

3. **Simplify both sides:**
   \[
   x^2 - 3x + \frac{9}{4} = -\frac{32}{4} + \frac{9}{4}
   \]
   \[
   x^2 - 3x + \frac{9}{4} = -\frac{23}{4}
   \]

4. **Rewrite the left side as a square:** The left side of the equation is now a perfect square trinomial.
   \[
   \left(x - \frac{3}{2}\right)^2 = -\frac{23}{4}
   \]

5. **Solve for \(x\):** Take the square root of both sides.
   \[
   x - \frac{3}{2} = \pm \sqrt{-\frac{23}{4}}
   \]
   \[
   x - \frac{3}{2} = \pm \frac{\sqrt{23}i}{2}
   \]

6. **Isolate \(x\):**
   \[
   x = \frac{3}{2} \pm \frac{\sqrt{23}i}{2}
   \]
   \[
   x = \frac{3 + \sqrt{23}i}{2} \quad \text{or} \quad x = \frac{3 - \sqrt{23}i}{2}
   \]

So, the solutions are:
\[
x = \frac{3 + \sqrt{23}i}{2}
\]
\[
x = \frac{3 - \sqrt{23}i}{2}
\]

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. How do you recognize when to use the completing the square method?
2. Can completing the square be used for any quadratic equation?
3. How do you derive the quadratic formula from completing the square?
4. What are the advantages and disadvantages of completing the square compared to using the quadratic formula?
5. How do you graph a quadratic equation after completing the square?
6. What role do complex numbers play in completing the square for equations with negative discriminants?
7. How can you apply completing the square to solve quadratic equations in real-world problems?
8. What are some historical origins and applications of completing the square in mathematics?

**Tip:** Completing the square is a valuable technique not only for solving quadratic equations but also for understanding the derivation of the quadratic formula and analyzing the properties of parabolas.

Learn how to solve the quadratic equation x^2 - 3x + 8 using the completing the square method. This detailed step-by-step guide covers key concepts like quadratic equations, completing the square, and complex numbers. Suitable for students in Grades 10-12.

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