The equation displayed in the image is:

$-2x^2 + 4x - 7 = 0$

The task asks to solve this equation by completing the square.

Step-by-step solution:

1. Divide by -2 to simplify the equation: $x^2 - 2x + \frac{7}{2} = 0$

2. Move the constant term to the other side: $x^2 - 2x = -\frac{7}{2}$

3. Complete the square:

   • Take half the coefficient of $x$ (which is $-2$), divide by 2, and square it: $\left(\frac{-2}{2}\right)^2 = 1$
   • Add 1 to both sides: $x^2 - 2x + 1 = -\frac{7}{2} + 1$ $x^2 - 2x + 1 = -\frac{7}{2} + \frac{2}{2} = -\frac{5}{2}$

4. Factor the left-hand side: $(x - 1)^2 = -\frac{5}{2}$

5. Solve for $x$ by taking the square root of both sides: $x - 1 = \pm \sqrt{-\frac{5}{2}}$ $x - 1 = \pm i \sqrt{\frac{5}{2}}$

6. Isolate $x$: $x = 1 \pm i \sqrt{\frac{5}{2}}$

Thus, the solutions are:

$x = 1 + i \sqrt{\frac{5}{2}}, \quad x = 1 - i \sqrt{\frac{5}{2}}$

These are complex solutions.

Would you like to go over any specific details or ask further questions? Here are 5 related questions you might find helpful:

1. How do we handle complex numbers when completing the square?
2. What is the geometric interpretation of completing the square?
3. Can completing the square always be used to solve any quadratic equation?
4. How can we apply the quadratic formula to this equation instead?
5. What are the real-world applications of solving quadratic equations with complex solutions?

Tip: Always check if you can simplify the equation by dividing by a constant before completing the square—it makes the process easier!