What kind of Conic Section is 0^2-4(-4)(5)

Let's evaluate the expression \( 0^2 - 4(-4)(5) \) and then interpret the result in the context of conic sections.

### Step 1: Evaluate the expression
\[
0^2 - 4(-4)(5) = 0 + 80 = 80
\]

### Step 2: Interpretation
This expression can be considered the discriminant (\(\Delta\)) for a general quadratic equation of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), where:
\[
\Delta = B^2 - 4AC
\]

Given that \(\Delta = 80\) (which is greater than 0), we can determine the type of conic section:

- If \(\Delta > 0\), the conic section is a **hyperbola**.
- If \(\Delta = 0\), it is a **parabola**.
- If \(\Delta < 0\), it is an **ellipse** (or a circle in a special case).

Since \(\Delta = 80\), the conic section associated with this discriminant is a **hyperbola**.

### Do you want more details or have any questions?
Here are 5 relative questions:
1. How does the value of the discriminant affect the orientation of a hyperbola?
2. What are the key differences between hyperbolas and ellipses in terms of their equations?
3. How can you determine the asymptotes of a hyperbola from its equation?
4. What role does the term \(Bxy\) play in rotating a conic section?
5. How can you classify a conic section if the equation involves complex numbers?

**Tip**: The discriminant helps in classifying conic sections easily, especially when dealing with general quadratic equations.

Learn how to evaluate a quadratic discriminant and determine the type of conic section it represents. This problem explores key algebraic concepts and conic section properties. Suitable for high school students.

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