To solve the given equation and express it in a more familiar form, we can complete the square for both the \( x \) and \( y \) terms.

Given equation:
\[
x^2 - 6x + 9 + y^2 + 24y + 144 = 1
\]

1. **Group the \( x \) terms together**:
   \[
   (x^2 - 6x + 9)
   \]

2. **Group the \( y \) terms together**:
   \[
   (y^2 + 24y + 144)
   \]

3. **Notice that both groups are already perfect squares**:
   \[
   (x - 3)^2 + (y + 12)^2 = 1
   \]

So, the equation can be rewritten as:
\[
(x - 3)^2 + (y + 12)^2 = 1
\]

### Interpretation:
This is the equation of a circle with center at \( (3, -12) \) and radius \( 1 \).

Would you like more details on completing the square or any other related concepts? Here are some questions for further exploration:

1. How do you complete the square for more complex quadratic expressions?
2. What are the general forms of the equations for circles, ellipses, and hyperbolas?
3. How does the equation of a circle change when you shift its center?
4. Can you generalize the method for any quadratic equation in two variables?
5. How can this method be applied to find the vertex of a parabola?

**Tip:** Completing the square is a powerful tool not only for solving quadratic equations but also for transforming them into standard forms, making it easier to identify key features like vertices, centers, and radii.

Learn how to solve the equation x² - 6x + 9 + y² + 24y + 144 = 1 by completing the square. This problem involves key algebraic concepts, including completing the square and the equation of a circle. Suitable for students in Grades 9-10.

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