https://file.aiquickdraw.com/mathbot/file/1728569968471nvhg383o.jpg

The equation shown in the image is:

\[
x^2 + y^2 - 6x - 2y - 6 = 0
\]

To analyze or solve this, we can interpret it as the equation of a circle in its general form. We can complete the square for both \(x\) and \(y\) to rewrite it in standard form.

### Steps to complete the square:

1. Group the \(x\)-terms and \(y\)-terms:
   \[
   (x^2 - 6x) + (y^2 - 2y) = 6
   \]

2. Complete the square for \(x^2 - 6x\):
   \[
   x^2 - 6x = (x - 3)^2 - 9
   \]

3. Complete the square for \(y^2 - 2y\):
   \[
   y^2 - 2y = (y - 1)^2 - 1
   \]

4. Substitute these into the equation:
   \[
   (x - 3)^2 - 9 + (y - 1)^2 - 1 = 6
   \]

5. Simplify:
   \[
   (x - 3)^2 + (y - 1)^2 - 10 = 6
   \]

6. Add 10 to both sides:
   \[
   (x - 3)^2 + (y - 1)^2 = 16
   \]

This is the standard form of the equation of a circle:
\[
(x - 3)^2 + (y - 1)^2 = 16
\]

### Interpretation:
- The center of the circle is at \( (3, 1) \).
- The radius of the circle is \( \sqrt{16} = 4 \).

Would you like more details or have any questions?

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Here are 5 related questions that expand on this topic:

1. How do you derive the equation of a circle from two given points?
2. What is the significance of completing the square in conic sections?
3. How do you find the equation of a tangent line to a circle at a specific point?
4. How can you determine the intersection points of two circles algebraically?
5. What is the general formula for the equation of a circle in 3D space?

**Tip:** When working with circles, completing the square is a key tool to convert general equations into standard form for easier interpretation.

Solve the equation: x^2 + y^2 - 6x - 2y - 6 = 0

Solve the equation x^2 + y^2 - 6x - 2y - 6 = 0 by completing the square and converting it into the standard form of a circle. Learn how to find the center and radius of the circle through this detailed, step-by-step solution. Ideal for Grades 10-12.

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