Quadratic Inequalities and the Discriminant

A set of vocabulary flashcards reviewing the discriminant, its graphical interpretations, and solving quadratic inequalities based on lecture notes.

Discriminant

The mathematical expression $b^2 - 4ac$ used to determine the number and type of roots of a quadratic equation.

Discriminant Summary Diagram

A visual guide illustrating the three discriminant cases for a quadratic curve: $b^2 - 4ac > 0$ ($2$ distinct roots / $2$ POI), $b^2 - 4ac = 0$ ($2$ equal roots / $1$ POI tangent), and $b^2 - 4ac < 0$ (no real roots / no POI above $x$-axis).

Two Distinct Roots Condition

The condition $b^2 - 4ac > 0$, where the quadratic graph intersects the $x$-axis at $2$ distinct points of intersection ($2$ POI).

Two Equal Roots Condition

The condition $b^2 - 4ac = 0$, where the quadratic graph touches the $x$-axis at $1$ point of intersection ($1$ POI) as a tangent.

No Real Roots Condition

The condition $b^2 - 4ac < 0$, where the quadratic graph has no points of intersection (no POI) with the $x$-axis and remains strictly above the $x$-axis for a positive parabola.

Critical Values (Roots) of a Quadratic Inequality

The values obtained by setting the factored quadratic expression equal to $0$ (e.g., $-2$ and $6$ from $(k + 2)(k - 6) > 0$), which define the boundary points for the solution regions of the inequality.

Discriminant

The mathematical expression $b^2 - 4ac$ used to analyze the roots and intersections of a quadratic equation.

Graphical classification of the discriminant

A visual guide illustrating the three discriminant conditions: $b^2 - 4ac > 0$ (2 POI), $b^2 - 4ac = 0$ (1 POI / Tangent), and $b^2 - 4ac < 0$ (No POI / Above x-axis).

Discriminant condition $b^2 - 4ac > 0$

Indicates two distinct roots and 2 points of intersection (2 POI) with the x-axis.

Discriminant condition $b^2 - 4ac = 0$

Indicates two equal roots, 1 point of intersection (1 POI), and a tangent to the x-axis.

Discriminant condition $b^2 - 4ac < 0$

Indicates no real roots, no points of intersection (No POI), and a graph that remains above the x-axis for a positive parabola.

Critical values of $(k + 2)(k - 6) > 0$

The roots $-2$ and $6$ that define the boundaries of the quadratic inequality.

Solution set of $(k + 2)(k - 6) > 0$

The inequality solution $k < -2$ or $k > 6$, representing the outer tails above the x-axis for a positive parabola.