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To solve the equation \( 2x^2 + 8x = 9 \) by completing the square, let's go through the steps in detail:

1. **Divide by 2** (to make the coefficient of \( x^2 \) equal to 1):
   \[
   x^2 + 4x = \frac{9}{2}
   \]

2. **Complete the square**:
   - Take half of the coefficient of \( x \) (which is 4), square it, and add it to both sides.
   - Half of 4 is 2, and \( 2^2 = 4 \).
   - Add 4 to both sides of the equation:
     \[
     x^2 + 4x + 4 = \frac{9}{2} + 4
     \]
   - Simplify the right side:
     \[
     x^2 + 4x + 4 = \frac{9}{2} + \frac{8}{2} = \frac{17}{2}
     \]

3. **Rewrite as a square**:
   - The left side is a perfect square trinomial:
     \[
     (x + 2)^2 = \frac{17}{2}
     \]

4. **Solve for \( x \)** by taking the square root of both sides:
   \[
   x + 2 = \pm \sqrt{\frac{17}{2}}
   \]
   - Rewrite \( \sqrt{\frac{17}{2}} \) as \( \frac{\sqrt{17}}{\sqrt{2}} \), and then rationalize the denominator:
     \[
     x + 2 = \pm \frac{\sqrt{34}}{2}
     \]

5. **Isolate \( x \)**:
   \[
   x = -2 \pm \frac{\sqrt{34}}{2}
   \]

So, the solution set is:
\[
x = -2 + \frac{\sqrt{34}}{2}, \quad x = -2 - \frac{\sqrt{34}}{2}
\]

Let me know if you need further details on any step!

### Related Questions
1. How do you complete the square for other types of quadratic equations?
2. What if the equation was in the form \( ax^2 + bx + c = 0 \) without dividing by \( a \)?
3. How do we use completing the square to find the vertex of a quadratic function?
4. What are some practical applications of solving quadratic equations?
5. How is completing the square used in calculus, such as for integration?

**Tip:** Completing the square is especially helpful for converting a quadratic equation into vertex form, which reveals key features like the vertex of a parabola.

Solve the equation by completing the square: 2x^2 + 8x = 9

Learn how to solve the quadratic equation 2x^2 + 8x = 9 by completing the square. This step-by-step guide demonstrates key algebraic concepts and provides an exact solution using radicals. Suitable for high school students in Grades 9-12.

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