To solve the equation \( 2x^2 + 2y^2 - 20x - 20y + 50 = 0 \), we will first rewrite it in a more familiar form by completing the square for both \(x\) and \(y\).

### Step-by-step process:

1. **Divide the whole equation by 2** to simplify:
   \[
   x^2 + y^2 - 10x - 10y + 25 = 0
   \]

2. **Rearrange the equation** grouping terms involving \(x\) and \(y\):
   \[
   (x^2 - 10x) + (y^2 - 10y) = -25
   \]

3. **Complete the square** for both \(x\) and \(y\).

   - For the \(x\)-terms: 
     \[
     x^2 - 10x = (x - 5)^2 - 25
     \]
   
   - For the \(y\)-terms: 
     \[
     y^2 - 10y = (y - 5)^2 - 25
     \]

4. **Substitute the completed squares back into the equation**:
   \[
   (x - 5)^2 - 25 + (y - 5)^2 - 25 = -25
   \]

5. Simplify:
   \[
   (x - 5)^2 + (y - 5)^2 = 25
   \]

### Final Answer:
This is the equation of a circle with center at \( (5, 5) \) and radius \( 5 \).

Would you like more details or have any questions? Here are some related questions:

1. How do you derive the equation of a circle from a general quadratic form?
2. Can you explain why completing the square is useful in this context?
3. What is the geometric interpretation of the final equation?
4. How would you graph the circle based on this equation?
5. How do we find the radius from the standard form of a circle?

**Tip:** Completing the square is a valuable technique for converting quadratic equations into standard geometric forms, such as circles or parabolas.

Explore the process of converting the quadratic equation 2x^2 + 2y^2 - 20x - 20y + 50 = 0 into the standard equation of a circle. This solution covers key algebraic techniques such as completing the square, relevant for students in Grades 9-11.

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