https://file.aiquickdraw.com/mathbot/file/17264841882894an3nvq3.jpg

The equation provided is:

\[
x^2 + y^2 - 10x - 6y + 25 = 0
\]

This equation is a form of a circle equation. To rewrite it in standard form, we will need to complete the square for both \(x\) and \(y\).

### Step 1: Group the \(x\)-terms and \(y\)-terms

\[
(x^2 - 10x) + (y^2 - 6y) = -25
\]

### Step 2: Complete the square for \(x\)-terms

For \(x^2 - 10x\), take half of \(-10\), square it, and add it inside the parentheses:
\[
\frac{-10}{2} = -5, \quad (-5)^2 = 25
\]

So, we add 25 to both sides.

\[
(x^2 - 10x + 25) + (y^2 - 6y) = -25 + 25
\]

### Step 3: Complete the square for \(y\)-terms

For \(y^2 - 6y\), take half of \(-6\), square it, and add it inside the parentheses:
\[
\frac{-6}{2} = -3, \quad (-3)^2 = 9
\]

So, we add 9 to both sides.

\[
(x^2 - 10x + 25) + (y^2 - 6y + 9) = 0 + 9
\]

### Step 4: Rewrite the expression

Now, both expressions are perfect squares:
\[
(x - 5)^2 + (y - 3)^2 = 9
\]

This is the equation of a circle in standard form, where the center is \((5, 3)\) and the radius is \(\sqrt{9} = 3\).

### Final Answer:
The equation represents a circle with:
- **Center**: \((5, 3)\)
- **Radius**: 3

Would you like more details or have any further questions?

Here are 5 related questions:
1. How do you complete the square for a quadratic expression?
2. What is the general form of a circle's equation?
3. How do you find the radius and center from a circle's equation?
4. What are other methods to identify conic sections?
5. Can this equation represent anything other than a circle?

**Tip**: Always ensure to add the same number to both sides of the equation when completing the square!

Learn how to solve the equation x^2 + y^2 - 10x - 6y + 25 = 0 by completing the square to find the standard form of a circle's equation. The solution covers key algebraic concepts and circle properties, suitable for students in Grades 9-11.

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