key term - Discriminant

Definition

The discriminant is a mathematical expression that determines the nature of the solutions to a quadratic equation. It plays a crucial role in understanding the behavior and characteristics of polynomial equations, quadratic equations, and their graphical representations.

5 Must Know Facts For Your Next Test

1. The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is defined as $b^2 - 4ac$.
2. The discriminant determines the nature and number of real roots of a 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 (only complex roots).

Review Questions

• Explain how the discriminant is used to solve quadratic equations using the quadratic formula.
  • The discriminant, $b^2 - 4ac$, is a key component of the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The sign and value of the discriminant determine the nature of the roots of the quadratic equation. If the discriminant is positive, the equation has two distinct real roots; if it is zero, the equation has one real root (a repeated root); and if it is negative, the equation has no real roots, only complex roots. Understanding the discriminant is essential for correctly applying the quadratic formula to solve quadratic equations.
• Describe how the discriminant is used to graph quadratic functions and determine their key features.
  • The discriminant provides important information about the graph of a quadratic function, $f(x) = ax^2 + bx + c$. If the discriminant, $b^2 - 4ac$, is positive, the graph is a parabola that opens upward or downward and intersects the $x$-axis at two distinct points (the roots). If the discriminant is zero, the graph is a parabola that opens upward or downward and has one point of intersection with the $x$-axis (a repeated root). If the discriminant is negative, the graph is a parabola that opens upward or downward and does not intersect the $x$-axis (no real roots). The discriminant also helps determine the location of the vertex, which is the point of maximum or minimum value of the function.
• Analyze how the discriminant can be used to solve quadratic inequalities and understand the behavior of the solutions.
  • The discriminant plays a crucial role in solving quadratic inequalities of the form $ax^2 + bx + c \geq 0$ or $ax^2 + bx + c \leq 0$. The sign of the discriminant, $b^2 - 4ac$, determines the number and nature of the solutions. If the discriminant is positive, the inequality has two real solutions, and the graph of the quadratic function is a parabola that opens upward or downward and intersects the $x$-axis at two points. If the discriminant is zero, the inequality has one real solution, and the graph is a parabola that opens upward or downward and touches the $x$-axis at a single point. If the discriminant is negative, the inequality has no real solutions, and the graph of the quadratic function is a parabola that opens upward or downward and does not intersect the $x$-axis. Understanding the discriminant is essential for correctly solving and interpreting the solutions to quadratic inequalities.

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 $x$ that satisfy a polynomial equation, making the equation equal to zero.

Vertex:

The point on a parabolic graph where the function changes from increasing to decreasing or vice versa.