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!