To find the center and radius of the circle represented by the equation \( x^2 + y^2 + 6x - 18y + 23 = 0 \), we need to rewrite the equation in the standard form of a circle equation, which is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center and \(r\) is the radius.

### Steps:

1. **Group the x and y terms:**
   \[
   (x^2 + 6x) + (y^2 - 18y) = -23
   \]

2. **Complete the square for both \(x\) and \(y\):**

   - For \(x^2 + 6x\):
     - Take half the coefficient of \(x\), which is 6, divide by 2 to get 3, and then square it: \(3^2 = 9\).
     - Add and subtract 9.

   - For \(y^2 - 18y\):
     - Take half the coefficient of \(y\), which is -18, divide by 2 to get -9, and then square it: \((-9)^2 = 81\).
     - Add and subtract 81.

   The equation becomes:
   \[
   (x^2 + 6x + 9) + (y^2 - 18y + 81) = -23 + 9 + 81
   \]

3. **Simplify and rewrite:**
   \[
   (x + 3)^2 + (y - 9)^2 = 67
   \]

### Conclusion:

The standard form of the circle's equation is:

\[
(x + 3)^2 + (y - 9)^2 = 67
\]

- **Center**: \((-3, 9)\)
- **Radius**: \(\sqrt{67}\)

Would you like more details on any of these steps or further explanation?

### Related Questions:
1. What is the general form of a circle equation?
2. How do you complete the square for a quadratic expression?
3. What is the significance of the radius in a circle equation?
4. Can you always convert a quadratic equation into the standard circle form?
5. How does translating the center of the circle affect the equation?
6. What are some examples of circle equations in real-world applications?
7. How do you derive the formula for completing the square?
8. What is the geometric interpretation of the center and radius of a circle?

**Tip:** Completing the square is a valuable technique not only for circles but also for solving quadratic equations and analyzing parabolas.

Learn how to find the center and radius of a circle given the equation x^2 + y^2 + 6x - 18y + 23 = 0. This problem involves transforming the circle equation into standard form using completing the square method. Suitable for high school students studying algebra and geometry.

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