Discriminant

Definition

The discriminant is a value that determines the nature of the solutions to a quadratic equation. It provides information about the number and type of real roots that the equation has.

5 Must Know Facts For Your Next Test

1. The discriminant is calculated using the formula $b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.
2. The sign of the discriminant determines the nature of the roots of the quadratic equation.
3. If the discriminant is positive, the equation has two distinct real roots.
4. If the discriminant is zero, the equation has one real root (a repeated root).
5. If the discriminant is negative, the equation has no real roots, but rather two complex conjugate roots.

Review Questions

• Explain how the discriminant is used to determine the number and type of real roots for a quadratic equation.
  • The discriminant, calculated as $b^2 - 4ac$, is used to determine the number and type of real roots for a quadratic equation. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root (a repeated root). If the discriminant is negative, the equation has no real roots, but rather two complex conjugate roots.
• Describe the relationship between the discriminant and the solutions to a quadratic equation.
  • The discriminant is directly related to the solutions of a quadratic equation. A positive discriminant means the equation has two distinct real roots, a zero discriminant means the equation has one real root (a repeated root), and a negative discriminant means the equation has no real roots, but rather two complex conjugate roots. The value of the discriminant provides information about the nature and number of solutions to the quadratic equation.
• Analyze how the discriminant can be used to classify the different types of quadratic functions.
  • The discriminant can be used to classify quadratic functions into different categories based on the nature of their roots. If the discriminant is positive, the quadratic function has two distinct real roots and is considered a parabola with two intersections with the $x$-axis. If the discriminant is zero, the quadratic function has one real root and is considered a parabola with one intersection with the $x$-axis. If the discriminant is negative, the quadratic function has no real roots and is considered a parabola that does not intersect the $x$-axis, but rather has two complex conjugate roots.

Related terms

Quadratic Equation:

A polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

Real Roots: The values of $x$ that satisfy the quadratic equation and make the equation equal to zero.

Complex Roots:

The solutions to a quadratic equation that are not real numbers, but rather involve imaginary components.