It looks like you've uploaded a file with an image. I'll extract the mathematical content from it and provide a solution. Please give me a moment. PIL import Image

Open the uploaded image to examine its content

img_path = '/mnt/data/173351936967374al4sia.jpg' image = Image.open(img_path)

Show the image to understand its content

image.show()I have opened the image, but it appears I need to extract the mathematical details from it. Let me process the image further to analyze its content. pytesseract

Use OCR to extract text from the image

extracted_text = pytesseract.image_to_string(image)

Display the extracted text

extracted_textThe image contains the equation to be solved:

$2x^2 + 8x - 9 = 0$

We are asked to solve for all values of $x$ by completing the square and express the answer in the simplest form.

Let me proceed with solving it.Let's solve the quadratic equation $2x^2 + 8x - 9 = 0$ by completing the square:

Step 1: Move the constant term to the other side

$2x^2 + 8x = 9$

Step 2: Divide through by the coefficient of $x^2$, which is 2

$x^2 + 4x = \frac{9}{2}$

Step 3: Complete the square

To 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 $2^2 = 4$.

$x^2 + 4x + 4 = \frac{9}{2} + 4$

Step 4: Simplify the right-hand side

$x^2 + 4x + 4 = \frac{9}{2} + \frac{8}{2} = \frac{17}{2}$

Step 5: Rewrite the left side as a perfect square

$(x + 2)^2 = \frac{17}{2}$

Step 6: Take the square root of both sides

$x + 2 = \pm \sqrt{\frac{17}{2}}$

Step 7: Simplify the square root

$x + 2 = \pm \frac{\sqrt{34}}{2}$

Step 8: Solve for $x$

$x = -2 \pm \frac{\sqrt{34}}{2}$

Final Answer:

$x = -2 + \frac{\sqrt{34}}{2} \quad \text{or} \quad x = -2 - \frac{\sqrt{34}}{2}$

This is the solution to the equation in its simplest form.

Would you like further clarification or details on any step? Here are some related questions you may find useful:

1. How does completing the square work for equations where the coefficient of $x^2$ is not 1?
2. What is the general process for solving quadratic equations using the quadratic formula?
3. How can you check the solutions to a quadratic equation by substitution?
4. What is the geometric interpretation of completing the square in a quadratic equation?
5. How do you solve a quadratic equation by factoring instead of completing the square?

Tip: When completing the square, always remember to adjust both sides of the equation to maintain equality after adding a constant to the left side.