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The question asks to find the radius and center of the circle from the equation provided, using the method of completing the square.

The equation given is:

\[
x^2 + y^2 + 4x - 6y + 12 = 0
\]

### Steps to find the center and radius:

1. **Rearrange the equation**:
   \[
   x^2 + 4x + y^2 - 6y = -12
   \]
   Move the constant to the other side of the equation.

2. **Complete the square**:
   - For the **x** terms: \( x^2 + 4x \)
     - Take half of 4 (which is 2), square it to get \( 2^2 = 4 \). Add 4 to both sides.
   - For the **y** terms: \( y^2 - 6y \)
     - Take half of -6 (which is -3), square it to get \( (-3)^2 = 9 \). Add 9 to both sides.
   
   Adding 4 and 9 to both sides gives:
   \[
   (x^2 + 4x + 4) + (y^2 - 6y + 9) = -12 + 4 + 9
   \]
   Simplifies to:
   \[
   (x + 2)^2 + (y - 3)^2 = 1
   \]

3. **Identify the center and radius**:
   - The equation is now in the standard form of a circle: \( (x - h)^2 + (y - k)^2 = r^2 \)
   - The center is \( (-2, 3) \) and the radius is \( \sqrt{1} = 1 \).

### Final answer:
- **Center**: \( (-2, 3) \)
- **Radius**: \( 1 \)

Would you like further details on completing the square, or have any questions?

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Here are 5 related questions to expand your understanding:

1. How do you identify the general form of a circle equation?
2. Why do we add and subtract the same value when completing the square?
3. What is the significance of the center and radius in the geometry of a circle?
4. How would the equation change if the radius was larger, say 5?
5. Can completing the square be used for any quadratic equation, and how?

**Tip**: Always verify your final equation by expanding the completed square form to check if it matches the original equation.

Find the radius and the center of the circle from the equation x^2 + y^2 + 4x - 6y + 12 = 0 using completing the square.

Learn how to find the radius and center of a circle from the equation x^2 + y^2 + 4x - 6y + 12 = 0 using the completing the square method. This step-by-step solution explains the key algebraic concepts and formulas involved. Suitable for students in Grades 9-12.

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