To solve the equation \( x^2 - 8x - 8y - 24 = 0 \), we can rewrite it in a more familiar form. This equation is a conic, and it looks like a parabola. Let's solve for \( y \) and rewrite it in vertex form.

### Step 1: Rearrange the equation to isolate \( y \)
\[
x^2 - 8x - 24 = 8y
\]
\[
y = \frac{x^2 - 8x - 24}{8}
\]

### Step 2: Complete the square for the expression involving \( x \)
The expression \( x^2 - 8x \) can be completed to a square. First, take half of the coefficient of \( x \) (which is \(-8\)) and square it:
\[
\left( \frac{-8}{2} \right)^2 = 16
\]
Now, add and subtract \( 16 \) inside the equation to complete the square:
\[
x^2 - 8x = (x - 4)^2 - 16
\]

### Step 3: Substitute this back into the equation
\[
y = \frac{(x - 4)^2 - 16 - 24}{8}
\]
Simplify the constants:
\[
y = \frac{(x - 4)^2 - 40}{8}
\]
\[
y = \frac{(x - 4)^2}{8} - 5
\]

### Final result:
The equation in vertex form is:
\[
y = \frac{1}{8}(x - 4)^2 - 5
\]
This is the equation of a parabola with vertex at \( (4, -5) \) and it opens upwards since the coefficient of \( (x - 4)^2 \) is positive.

Would you like more details, or do you have any questions?

### Related Questions:
1. How can we determine the axis of symmetry for this parabola?
2. What is the focus and directrix of this parabola?
3. How does the coefficient \( \frac{1}{8} \) affect the shape of the parabola?
4. Can you derive the equation of the parabola from the general conic form?
5. What other methods can be used to rewrite a quadratic equation?

### Tip:
When completing the square, always remember to adjust for any constants outside the square to maintain the equation's balance.

Learn how to solve the quadratic equation x² - 8x - 8y - 24 = 0 by completing the square. This step-by-step solution explains how to rewrite the equation in vertex form, identify the vertex, and determine the shape of the parabola. Ideal for students in Grades 9-11.

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