(X+4)^2+(y-4)^2=25 find the general equation to standard equation

The equation provided is already in the standard form of a circle:

\[
(x + 4)^2 + (y - 4)^2 = 25
\]

### Steps to convert it to the general form:

1. **Expand the squared terms**:
   - Expand \((x + 4)^2\):
     \[
     (x + 4)^2 = x^2 + 8x + 16
     \]
   - Expand \((y - 4)^2\):
     \[
     (y - 4)^2 = y^2 - 8y + 16
     \]

2. **Substitute these expansions back into the equation**:
   \[
   x^2 + 8x + 16 + y^2 - 8y + 16 = 25
   \]

3. **Combine like terms**:
   \[
   x^2 + y^2 + 8x - 8y + 32 = 25
   \]

4. **Move the constant term to the other side to obtain the general form**:
   \[
   x^2 + y^2 + 8x - 8y + 7 = 0
   \]

### General form of the equation:
\[
x^2 + y^2 + 8x - 8y + 7 = 0
\]

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Would you like any more details or have any questions?

Here are some related questions:

1. How do you convert a general form of a circle equation to the standard form?
2. What is the geometric interpretation of the constants in the general form equation?
3. How does the radius of the circle appear in the general form equation?
4. What are the steps to find the center of the circle from the general equation?
5. How can the general equation be used to determine if a point lies on the circle?
6. What happens if the equation's constant term is negative?
7. How would the equation change if the circle's radius were larger?
8. How do you derive the equation of a circle with a different center?

**Tip:** Always re-check your expansions when

Learn how to convert the circle equation (X+4)^2+(y-4)^2=25 to its general form step-by-step. Understand the geometric interpretation of circle equations and the steps involved in converting between standard and general forms. Suitable for high school students studying geometry and circle equations.

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