To solve the equation \( x^2 - 4x - 8y - 4 = 0 \), we can rewrite it in the form of a standard conic section equation. 

### Step 1: Rearrange the equation
First, rearrange the equation to group the \(x\) and \(y\) terms separately:

\[
x^2 - 4x = 8y + 4
\]

### Step 2: Complete the square for the \(x\) terms
To complete the square for \(x^2 - 4x\), follow these steps:

1. Take the coefficient of \(x\) (which is \(-4\)), divide it by 2, and then square it:
   \[
   \left(\frac{-4}{2}\right)^2 = (-2)^2 = 4
   \]

2. Add and subtract this square on the left-hand side:
   \[
   x^2 - 4x + 4 - 4 = 8y + 4
   \]

3. Simplify the left-hand side:
   \[
   (x - 2)^2 - 4 = 8y + 4
   \]

### Step 3: Isolate the \(y\) term
Now, move the constant term to the right-hand side:

\[
(x - 2)^2 = 8y + 8
\]

### Step 4: Simplify the equation
Finally, divide everything by 8 to solve for \(y\):

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

or equivalently,

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

### Final Answer:
The equation represents a parabola that opens upwards. The vertex of the parabola is at \((2, -1)\), and it is in the form:

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

### Would you like more details or have any questions? 

Here are some related questions you might find interesting:

1. How do you determine the direction in which a parabola opens?
2. What is the vertex form of a parabola, and how do you derive it?
3. How do you complete the square for quadratic equations?
4. How does the coefficient of the squared term affect the shape of the parabola?
5. Can this equation be converted into a different form, such as standard form?
6. How do you find the focus and directrix of a parabola?
7. What are conic sections, and how are they classified?
8. How does the discriminant help determine the nature of a conic section?

**Tip:** Completing the square is a useful technique for converting a quadratic equation into a more manageable form, often revealing important features like the vertex of a parabola.

Learn how to solve the quadratic equation x^2 - 4x - 8y - 4 = 0 with a detailed step-by-step solution. This problem involves completing the square for quadratic equations, revealing the vertex form of a parabola. Suitable for high school students in Grades 10-12.

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