Discriminant

Definition

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

5 Must Know Facts For Your Next Test

1. The discriminant is defined as $b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.
2. The sign of the discriminant determines the nature of the roots of the quadratic equation: if the discriminant is positive, the equation has two real roots; if the discriminant is zero, the equation has one real root; if the discriminant is negative, the equation has two complex conjugate roots.
3. The discriminant is used to classify quadratic functions into three basic classes: parabolas that open upward, parabolas that open downward, and degenerate cases where the parabola is a line or a point.
4. The discriminant is also used to determine the vertex and the axis of symmetry of a quadratic function, which are important features in the analysis of the function's behavior.
5. Understanding the discriminant is crucial in solving and analyzing quadratic equations, which are fundamental in many areas of mathematics, science, and engineering.

Review Questions

• Explain how the sign of the discriminant determines the nature of the roots of a quadratic equation.
  • The sign of the discriminant, $b^2 - 4ac$, determines the nature of the roots of the quadratic equation $ax^2 + bx + c = 0$. If the discriminant is positive, the equation has two real and distinct roots. If the discriminant is zero, the equation has one real root (a repeated root). If the discriminant is negative, the equation has two complex conjugate roots. This information is crucial in understanding the behavior of quadratic functions and solving quadratic equations.
• Describe how the discriminant can be used to classify quadratic functions into three basic classes.
  • The discriminant can be used to classify quadratic functions into three basic classes: parabolas that open upward, parabolas that open downward, and degenerate cases where the parabola is a line or a point. If the discriminant is positive, the quadratic function is a parabola that opens upward or downward. If the discriminant is zero, the quadratic function is a line or a point. Understanding these classifications is essential in analyzing the properties and behavior of quadratic functions.
• Analyze how the discriminant can be used to determine the vertex and axis of symmetry of a quadratic function.
  • The discriminant can be used to determine the vertex and axis of symmetry of a quadratic function. The vertex of a quadratic function $f(x) = ax^2 + bx + c$ is given by $(-b/2a, f(-b/2a))$, and the axis of symmetry is the vertical line $x = -b/2a$. The discriminant $b^2 - 4ac$ is used to calculate these important features of the quadratic function, which provide valuable information about the function's behavior and can be used in various applications, such as optimization problems.

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.

Roots:

The values of the variable $x$ that satisfy the quadratic equation, i.e., the solutions to the equation.

Real Numbers: The set of all numbers that can be represented on the number line, including positive and negative integers, fractions, and irrational numbers.