Use the method of completing the square to write an equation of the circle in standard form.
x2 + y2 − 12x − 10y + 56 = 0

To rewrite the given equation of the circle \( x^2 + y^2 - 12x - 10y + 56 = 0 \) in standard form using the method of completing the square, follow these steps:

### Step 1: Rearrange the terms.
Group the \( x \)-terms and \( y \)-terms together and move the constant to the other side of the equation:
\[
x^2 - 12x + y^2 - 10y = -56
\]

### Step 2: Complete the square for the \( x \)-terms.
Take the coefficient of \( x \), divide it by 2, and then square it:
\[
\text{For } x^2 - 12x: \left(\frac{-12}{2}\right)^2 = (-6)^2 = 36
\]
Now add 36 to both sides of the equation:
\[
x^2 - 12x + 36 + y^2 - 10y = -56 + 36
\]
\[
(x - 6)^2 + y^2 - 10y = -20
\]

### Step 3: Complete the square for the \( y \)-terms.
Take the coefficient of \( y \), divide it by 2, and then square it:
\[
\text{For } y^2 - 10y: \left(\frac{-10}{2}\right)^2 = (-5)^2 = 25
\]
Now add 25 to both sides of the equation:
\[
(x - 6)^2 + (y - 5)^2 = -20 + 25
\]
\[
(x - 6)^2 + (y - 5)^2 = 5
\]

### Step 4: Write the equation in standard form.
The equation of the circle is now in standard form:
\[
(x - 6)^2 + (y - 5)^2 = 5
\]
This represents a circle with center \( (6, 5) \) and radius \( \sqrt{5} \).

Would you like more details or have any questions?

### Related questions:
1. How can you derive the general formula for completing the square?
2. How do you interpret the center and radius of a circle from its standard form?
3. Can you apply this method to a circle equation with different coefficients?
4. What are the geometric implications of completing the square?
5. How would this process change for an ellipse or hyperbola?

### Tip:
When completing the square, always remember to balance the equation by adding the same value to both sides to maintain equality.

Use the method of completing the square to write an equation of the circle in standard form. x^2 + y^2 − 12x − 10y + 56 = 0

Learn how to convert the circle equation x^2 + y^2 − 12x − 10y + 56 = 0 into its standard form using the method of completing the square. This problem covers key geometric concepts and completing the square steps. Suitable for students in Grades 9-11.

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