We are given the equation:

\[
8x^2 + 3y^2 - 24x + 6y - 32 = 0
\]

We will rewrite it in the form of the equation of an ellipse. This involves completing the square for both the \(x\)-terms and the \(y\)-terms.

### Step 1: Rearrange terms by grouping \(x\)-terms and \(y\)-terms:
\[
8x^2 - 24x + 3y^2 + 6y = 32
\]

### Step 2: Factor out the coefficients of \(x^2\) and \(y^2\) from their respective terms:
\[
8(x^2 - 3x) + 3(y^2 + 2y) = 32
\]

### Step 3: Complete the square:
For \(x^2 - 3x\), take half of \(-3\), square it, and add that inside the parentheses:
\[
x^2 - 3x = (x - \frac{3}{2})^2 - \frac{9}{4}
\]

For \(y^2 + 2y\), take half of \(2\), square it, and add that inside the parentheses:
\[
y^2 + 2y = (y + 1)^2 - 1
\]

Now substitute these back into the equation:

\[
8 \left( (x - \frac{3}{2})^2 - \frac{9}{4} \right) + 3 \left( (y + 1)^2 - 1 \right) = 32
\]

### Step 4: Simplify:
\[
8(x - \frac{3}{2})^2 - 18 + 3(y + 1)^2 - 3 = 32
\]

\[
8(x - \frac{3}{2})^2 + 3(y + 1)^2 = 53
\]

### Step 5: Divide by 53 to get the equation of an ellipse:
\[
\frac{(x - \frac{3}{2})^2}{\frac{53}{8}} + \frac{(y + 1)^2}{\frac{53}{3}} = 1
\]

Thus, the equation of the ellipse is:
\[
\frac{(x - \frac{3}{2})^2}{\frac{53}{8}} + \frac{(y + 1)^2}{\frac{53}{3}} = 1
\]

This represents an ellipse centered at \(\left(\frac{3}{2}, -1\right)\), with semi-major and semi-minor axes determined by the denominators.

Would you like more details or have any questions?

Here are five related questions:
1. How do you determine the center of an ellipse from its equation?
2. What is the general form of the equation of an ellipse?
3. How do you find the lengths of the semi-major and semi-minor axes?
4. How does completing the square help in transforming equations of conic sections?
5. How can you classify conic sections based on their equations?

**Tip**: For ellipses, the larger denominator corresponds to the semi-major axis length squared, and the smaller to the semi-minor.

Learn how to convert the equation 8x² + 3y² - 24x + 6y - 32 = 0 into the standard form of an ellipse using the method of completing the square. This detailed guide covers key mathematical concepts like conic sections and provides a step-by-step solution. Ideal for high school students in Grades 10-12.

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