Discriminant

Definition

The discriminant of a quadratic equation in the form $ax^2 + bx + c = 0$ is given by $\Delta = b^2 - 4ac$. It determines the nature and number of roots of the quadratic equation.

5 Must Know Facts For Your Next Test

1. The discriminant helps to classify conic sections: a positive discriminant indicates an ellipse or hyperbola, zero indicates a parabola, and negative indicates an imaginary ellipse or hyperbola.
2. In parametric equations, the discriminant can be used to determine if the resulting graph is an ellipse, parabola, or hyperbola.
3. For polar coordinates involving conic sections, the discriminant helps identify whether the conic section is open or closed.
4. When analyzing systems of quadratic equations, the discriminant provides insight into intersection points and their nature.
5. Understanding how to calculate and interpret the discriminant is crucial for solving problems involving conic sections in both Cartesian and polar coordinates.

Review Questions

• What does a negative discriminant indicate about the roots of a quadratic equation?
• How can you use the discriminant to classify a conic section?
• What is the formula for calculating the discriminant?

Related terms

Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$, where $a \neq 0$.

Conic Sections:

Curves obtained as intersections of a plane with a double-napped cone; includes ellipses, parabolas, and hyperbolas.

Parametric Equations:

Equations that express coordinates of points as functions of one or more independent parameters.