The given equation \((X-2)^2 + [Y-(-1)]^2 = 3^2\) represents a circle in its standard form:

\[
(X - h)^2 + (Y - k)^2 = r^2
\]

Where:
- The center of the circle \((h, k)\) is \((2, -1)\),
- The radius \(r\) is 3.

To convert this into the general form of a circle's equation:

\[
X^2 + Y^2 + DX + EY + F = 0
\]

we need to expand and simplify the equation.

### Step 1: Expand the equation

Expand \((X - 2)^2\) and \((Y + 1)^2\):

\[
(X - 2)^2 = X^2 - 4X + 4
\]
\[
(Y + 1)^2 = Y^2 + 2Y + 1
\]

So the equation becomes:

\[
X^2 - 4X + 4 + Y^2 + 2Y + 1 = 9
\]

### Step 2: Simplify the equation

Combine like terms:

\[
X^2 + Y^2 - 4X + 2Y + 5 = 9
\]

Now, move all terms to one side of the equation:

\[
X^2 + Y^2 - 4X + 2Y + 5 - 9 = 0
\]

This simplifies to:

\[
X^2 + Y^2 - 4X + 2Y - 4 = 0
\]

This is the general form of the given circle's equation.

Would you like more details or have any questions?

Here are some related questions:
1. How would the general equation change if the radius were doubled?
2. How can you determine the center and radius from a general form equation?
3. What would be the standard form of the equation \(X^2 + Y^2 - 6X + 8Y - 20 = 0\)?
4. How does completing the square help in converting from general to standard form?
5. What does the discriminant of the general equation tell us about the nature of the conic?

**Tip:** When expanding binomials like \((X - 2)^2\), remember the formula \((A - B)^2 = A^2 - 2AB + B^2\) to ensure accuracy in your calculations.

Explore the circle equation (X-2)^2 + [Y-(-1)]^2 = 3^2. Learn how to convert this equation from standard to general form step-by-step. Suitable for high school students studying circle equations and quadratic forms.

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