Page 2: Solving Quadratic Inequalities

This page delves into the process of solving quadratic inequalities, providing examples and explaining key concepts.

The page illustrates how to solve inequalities like x² - 4 < 0, demonstrating that the solution involves finding the roots and determining the intervals that satisfy the inequality.

Example: For x² - 4 < 0, the roots are x = -2 and x = 2, and the solution is -2 < x < 2.

It also introduces the concept of "external" and "internal" solutions, which depend on whether the inequality sign is > or <, and whether the parabola opens upward or downward.

Vocabulary:

• Soluzioni esterne: Solutions outside the roots
• Soluzioni interne: Solutions between the roots

The page emphasizes that finding the points where the parabola intersects the x-axis is crucial, and these points are found by solving the corresponding quadratic equation.

Highlight: The disequazioni di secondo grado studio del segno (sign analysis of quadratic inequalities) is essential for determining the solution intervals.

Page 3: Special Cases and Solution Methods

This page covers special cases in quadratic inequalities and provides a systematic approach to solving them.

It explains how to handle cases where the discriminant $Δ = b² - 4ac$ is zero, positive, or negative, affecting the nature of the roots.

Definition: The discriminant (Δ) determines the nature of the roots:

• Δ > 0: Two distinct real roots
• Δ = 0: One repeated real root
• Δ < 0: No real roots

The page provides a comprehensive schema disequazioni di secondo grado (scheme for quadratic inequalities), showing how to approach different cases based on the inequality sign and the parabola's orientation.

Example: For x² - 2x + 1 ≤ 0, we find that Δ = 0, leading to one repeated root at x = 1.

The page also includes a table summarizing the solutions for various scenarios, making it easier for students to understand and apply the concepts.

Highlight: The formula disequazioni secondo grado (formula for quadratic inequalities) depends on the sign of the leading coefficient and the nature of the roots.

This comprehensive guide provides students with the tools to solve various types of disequazioni di secondo grado esercizi (quadratic inequality exercises), including those involving disequazioni di secondo grado fratte (fractional quadratic inequalities).

Page 1: Introduction to Quadratic Inequalities

This page introduces the concept of quadratic inequalities and emphasizes the importance of understanding the parabola's position in solving them.

Definition: A quadratic inequality is an expression of the form ax² + bx + c ⋛ 0, where a, b, and c are constants, and a ≠ 0.

The page highlights that the parabola's orientation is crucial for solving disequazioni di secondo grado (quadratic inequalities). It explains that when the coefficient of x² is positive, the parabola opens upward, and when it's negative, the parabola opens downward.

Highlight: The direction of the parabola is determined by the sign of the coefficient of x².

The page also mentions that only the x-axis (abscissa) is used in solving these inequalities, simplifying the process by focusing on the roots and intervals along the x-axis.

Example: For the inequality x² - 2x > 0, we focus on where the parabola intersects the x-axis and the intervals between these points.