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To determine the center of the hyperbolic paraboloid given by the equation:

\[
4x^2 - 6y^2 - 8x + 24y - 24z - 20 = 0
\]

we will need to complete the square for both \(x\) and \(y\) terms and simplify the equation. Let's start solving this step by step.

### Step 1: Group the like terms

Group the \(x\)-terms, \(y\)-terms, and isolate the remaining terms.

\[
4(x^2 - 2x) - 6(y^2 - 4y) - 24z = 20
\]

### Step 2: Complete the square for \(x\) and \(y\)

**For \(x\):**

\[
x^2 - 2x = (x - 1)^2 - 1
\]

Thus, we can rewrite the \(x\)-terms as:

\[
4(x^2 - 2x) = 4((x - 1)^2 - 1) = 4(x - 1)^2 - 4
\]

**For \(y\):**

\[
y^2 - 4y = (y - 2)^2 - 4
\]

Thus, we can rewrite the \(y\)-terms as:

\[
-6(y^2 - 4y) = -6((y - 2)^2 - 4) = -6(y - 2)^2 + 24
\]

### Step 3: Substitute back into the equation

Substitute the completed square expressions back into the original equation:

\[
4(x - 1)^2 - 4 - 6(y - 2)^2 + 24 - 24z = 20
\]

Simplify:

\[
4(x - 1)^2 - 6(y - 2)^2 - 24z + 20 = 20
\]

Now, subtract 20 from both sides:

\[
4(x - 1)^2 - 6(y - 2)^2 - 24z = 0
\]

### Step 4: Find the center

From the final equation \(4(x - 1)^2 - 6(y - 2)^2 - 24z = 0\), it's clear that the center of the hyperbolic paraboloid is:

\[
(1, 2, 0)
\]

### Final Answer:
The center is \((1, 2, 0)\).

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Do you need further explanation on any of these steps?

Here are some related questions:
1. How do you complete the square for more complex equations?
2. What is the geometric interpretation of a hyperbolic paraboloid?
3. How do you classify a quadric surface based on its equation?
4. What are the properties of a hyperbolic paraboloid?
5. How would this problem change if the equation had a different constant term?

**Tip:** Completing the square is an essential technique when working with quadratic forms and can help in identifying the geometric nature of equations.

Determine the center of the hyperbolic paraboloid given by the equation 4x^2 - 6y^2 - 8x + 24y - 24z - 20 = 0.

This problem involves finding the center of a hyperbolic paraboloid by completing the square for the equation 4x^2 - 6y^2 - 8x + 24y - 24z - 20 = 0. The step-by-step guide covers algebraic methods such as completing the square and recognizing the standard form of quadratic surfaces, suitable for students in Grades 11-12.

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