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The problem in your image asks to express the equation \(x^2 + y^2 - 4x + 6y - 12 = 0\) in standard form. This is the equation of a circle, and to rewrite it in standard form, we need to complete the square for both \(x\) and \(y\).

1. Start with the given equation:
   \[
   x^2 + y^2 - 4x + 6y - 12 = 0
   \]

2. Rearrange the terms to group \(x\)-terms and \(y\)-terms:
   \[
   (x^2 - 4x) + (y^2 + 6y) = 12
   \]

3. Complete the square for \(x\):
   - Take half of the coefficient of \(x\) (which is -4), square it, and add it to both sides:
     \[
     \left( \frac{-4}{2} \right)^2 = 4
     \]
     So we add 4 to both sides.

4. Complete the square for \(y\):
   - Take half of the coefficient of \(y\) (which is 6), square it, and add it to both sides:
     \[
     \left( \frac{6}{2} \right)^2 = 9
     \]
     So we add 9 to both sides.

5. Now the equation becomes:
   \[
   (x^2 - 4x + 4) + (y^2 + 6y + 9) = 12 + 4 + 9
   \]
   Simplify:
   \[
   (x - 2)^2 + (y + 3)^2 = 25
   \]

This is the equation of a circle in standard form with center \((2, -3)\) and radius \(5\).

From the options in the image, the correct choice is:
\[
(x - 2)^2 + (y + 3)^2 = 25
\]

Do you need more details or have any further questions?

Here are five related questions to deepen your understanding:
1. How do you complete the square for other conic sections, such as ellipses?
2. Can you explain why completing the square works for any quadratic expression?
3. What would the standard form of a parabola look like, and how is it different from a circle?
4. How would you transform the equation if it represented a circle with a different center?
5. Can completing the square be used for higher-degree polynomials?

**Tip:** Always remember to keep both sides of the equation balanced when completing the square by adding the same value to both sides!

Express the equation x^2 + y^2 - 4x + 6y - 12 = 0 in standard form.

Discover how to convert the general equation of a circle x^2 + y^2 - 4x + 6y - 12 = 0 into its standard form by completing the square. This educational content is suitable for high school students learning advanced algebra and geometry.

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