Math Problem Statement

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

Solution

Let's evaluate the expression 024(4)(5)0^2 - 4(-4)(5) and then interpret the result in the context of conic sections.

Step 1: Evaluate the expression

024(4)(5)=0+80=800^2 - 4(-4)(5) = 0 + 80 = 80

Step 2: Interpretation

This expression can be considered the discriminant (Δ\Delta) for a general quadratic equation of the form Ax2+Bxy+Cy2+Dx+Ey+F=0Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, where: Δ=B24AC\Delta = B^2 - 4AC

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

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

Since Δ=80\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 BxyBxy 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.

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Discriminant
Conic Sections

Formulas

Quadratic equation discriminant formula

Theorems

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Suitable Grade Level

High School