The equation provided in the general form is:

$9x^2 - 4y^2 - 36x + 24y - 36 = 0$

We need to convert this to the standard form for hyperbolas, which looks like:

$\frac{(x - E)^2}{G} - \frac{(y - F)^2}{H} = 1$

Step 1: Rearranging and Grouping Terms

First, group the $x$ and $y$ terms together:

$9(x^2 - 4x) - 4(y^2 - 6y) = 36$

Step 2: Completing the Square

We will complete the square for both the $x$ and $y$ terms.

For the $x$ terms:

The expression inside the parentheses is $x^2 - 4x$. To complete the square, take half of $-4$ (which is $-2$), square it ($(-2)^2 = 4$), and add and subtract this value:

$9(x^2 - 4x + 4 - 4) = 9((x - 2)^2 - 4)$

This simplifies to:

$9(x - 2)^2 - 36$

For the $y$ terms:

The expression inside the parentheses is $y^2 - 6y$. Take half of $-6$ (which is $-3$), square it ($(-3)^2 = 9$), and add and subtract this value:

$-4(y^2 - 6y + 9 - 9) = -4((y - 3)^2 - 9)$

This simplifies to:

$-4(y - 3)^2 + 36$

Step 3: Substitute Back Into the Equation

Substitute these expressions back into the original equation:

$9(x - 2)^2 - 36 - 4(y - 3)^2 + 36 = 36$

Simplify:

$9(x - 2)^2 - 4(y - 3)^2 = 36$

Step 4: Divide by 36 to Normalize the Equation

To get the standard form, divide the entire equation by 36:

$\frac{9(x - 2)^2}{36} - \frac{4(y - 3)^2}{36} = \frac{36}{36}$

This simplifies to:

$\frac{(x - 2)^2}{4} - \frac{(y - 3)^2}{9} = 1$

Step 5: Identify the Constants

Now that we have the equation in standard form, we can identify the values of $E$, $F$, $G$, and $H$:

$E = 2, \quad F = 3, \quad G = 4, \quad H = 9$

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How do you complete the square for any quadratic expression?
2. What does the standard form of a hyperbola represent geometrically?
3. How can you distinguish between an ellipse and a hyperbola from the general equation?
4. Why is it necessary to divide the final equation by 36 in this problem?
5. What are the key characteristics of a hyperbola's asymptotes?

Tip: Completing the square is a key algebraic technique for converting quadratic expressions to vertex form. Always remember to add and subtract the same value inside the parentheses.