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.