https://file.aiquickdraw.com/mathbot/file/17255580486383vb0x3pi.jpg

To address this problem, let's begin by solving the questions in the image step-by-step. The given problem asks to convert the provided general equations of circles into their standard form.

### Standard Form of a Circle
The equation of a circle in standard form is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Where:
- \((h, k)\) is the center of the circle.
- \(r\) is the radius.

The general equation of a circle is typically given as:
\[
x^2 + y^2 + Dx + Ey + F = 0
\]
To convert this to standard form, we complete the square for both \(x\) and \(y\).

### Let's analyze each equation from the image:

---

#### (1) Equation A:
\[
x^2 + y^2 + 6x - 6y - 31 = 0
\]
1. Rearrange and group the terms:
   \[
   (x^2 + 6x) + (y^2 - 6y) = 31
   \]
2. Complete the square for both \(x\) and \(y\):
   - For \(x^2 + 6x\), add and subtract \((\frac{6}{2})^2 = 9\).
   - For \(y^2 - 6y\), add and subtract \((\frac{6}{2})^2 = 9\).
   \[
   (x^2 + 6x + 9) + (y^2 - 6y + 9) = 31 + 9 + 9
   \]
   \[
   (x + 3)^2 + (y - 3)^2 = 49
   \]
3. This is now in standard form:
   \[
   (x + 3)^2 + (y - 3)^2 = 7^2
   \]
   The center is \((-3, 3)\), and the radius is \(7\).

The answer corresponds to choice C.

---

#### (2) Equation B:
\[
x^2 + y^2 + 4x + 4y + 9 = 0
\]
1. Rearrange and group the terms:
   \[
   (x^2 + 4x) + (y^2 + 4y) = -9
   \]
2. Complete the square:
   - For \(x^2 + 4x\), add and subtract \((\frac{4}{2})^2 = 4\).
   - For \(y^2 + 4y\), add and subtract \((\frac{4}{2})^2 = 4\).
   \[
   (x^2 + 4x + 4) + (y^2 + 4y + 4) = -9 + 4 + 4
   \]
   \[
   (x + 2)^2 + (y + 2)^2 = -1
   \]
   This results in a negative value on the right side, meaning the equation does not represent a circle (this equation does not correspond to any valid circle).

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Would you like to continue with the rest of the problems or need further explanation on this part? 

Here are 5 related questions:
1. How do you derive the center and radius from the standard form equation of a circle?
2. What happens when the right side of the equation results in a negative value?
3. Why is completing the square necessary to convert

Use the information provided to write the standard form equations of each circle.

Learn how to convert the general form of circle equations to standard form by completing the square. This problem involves key algebraic concepts such as circle equations and completing the square, suitable for students in Grades 9-12.

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