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To solve the quadratic equation \(2x^2 + 9x - 5 = 0\) by completing the square, we need to follow these steps:

1. **Isolate the quadratic and linear terms**: Start by moving the constant term to the other side of the equation:
   \[
   2x^2 + 9x = 5
   \]

2. **Factor out the coefficient of \(x^2\)** from the left side:
   \[
   2(x^2 + \frac{9}{2}x) = 5
   \]

3. **Complete the square inside the parentheses**: To complete the square, take half the coefficient of \(x\), square it, and add it inside the parentheses. Since the coefficient of \(x\) is \(\frac{9}{2}\), half of this value is \(\frac{9}{4}\), and squaring \(\frac{9}{4}\) gives \(\frac{81}{16}\):
   \[
   2\left(x^2 + \frac{9}{2}x + \frac{81}{16}\right) = 5 + 2\left(\frac{81}{16}\right)
   \]
   Simplifying the right side:
   \[
   5 + \frac{162}{16} = \frac{80}{16} + \frac{162}{16} = \frac{242}{16} = \frac{121}{8}
   \]

4. **Write the equation in perfect square form** and solve for \(x\):
   \[
   2\left(x + \frac{9}{4}\right)^2 = \frac{121}{8}
   \]
   Dividing both sides by 2:
   \[
   \left(x + \frac{9}{4}\right)^2 = \frac{121}{16}
   \]
   Taking the square root of both sides:
   \[
   x + \frac{9}{4} = \pm\frac{11}{4}
   \]
   Solving for \(x\):
   \[
   x = \frac{11}{4} - \frac{9}{4} = \frac{2}{4} = \frac{1}{2} \quad \text{or} \quad x = -\frac{11}{4} - \frac{9}{4} = -\frac{20}{4} = -5
   \]

The answer is \(x = \frac{1}{2}\) or \(x = -5\), and the process to complete the square includes adding \(\frac{81}{16}\) to both sides. Therefore, the correct selection in the multiple-choice options you showed is:

- **Option D**. \(2x^2 + 9x - 5 = 0\) to \(x^2 + \frac{9}{2}x + \frac{81}{16} = \frac{121}{16}\)

Would you like to go over any of the steps in more detail or have any other questions?

Related questions you might consider:
1. How can you verify the solutions of a quadratic equation?
2. What are the advantages of completing the square compared to other methods like factoring or using the quadratic formula?
3. Can every quadratic equation be solved by completing the square?
4. How would the steps change if the coefficient of \(x^2\) was not 2 but another number?
5. What is the geometric interpretation of completing the square?

Tip: Always check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

Solve the quadratic equation by completing the square: 2x^2 + 9x - 5 = 0

This step-by-step solution explains how to solve the quadratic equation 2x^2 + 9x - 5 = 0 using the method of completing the square. Learn how to factor, complete the square, and find the roots of the equation. Suitable for students in Grades 9-12.

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