Practical Applications of Quadratic Inequalities

Objectives

1. Understand the definition and structure of quadratic inequalities.

2. Learn how to solve quadratic inequalities, paying attention to whether the coefficient 'a' is positive or negative.

3. Develop analytical skills and problem-solving abilities.

Contextualization

Quadratic inequalities are powerful mathematical tools that allow us to understand and solve problems where the solutions are not straightforward. Imagine, for example, predicting when a rocket reaches its maximum height or calculating the trajectory of a moving object. These real-world situations depend on the ability to solve quadratic inequalities, making this knowledge essential in various fields, including engineering, physics, and economics. For example, in civil engineering, quadratic inequalities are used to calculate material strength, while in finance they help evaluate the viability of investments.

Relevance of the Theme

Quadratic inequalities are extremely important in the current context due to their wide application in various fields. They are fundamental in solving complex problems in engineering, economics, and information technology, among others. Mastering the understanding and resolution of these inequalities prepares students to face technical and scientific challenges in the job market and in everyday life.

Definition of Quadratic Inequalities

Quadratic inequalities are mathematical expressions where the variable is raised to the second power. They have the general form ax² + bx + c > 0 (or < 0, ≤ 0, ≥ 0), where a, b, and c are real coefficients and a ≠ 0. They are used to determine intervals of values that satisfy the condition imposed by the inequality.

• The general form of a quadratic inequality is ax² + bx + c > 0.

• The coefficients a, b, and c are real numbers, and a must be different from zero.

• Quadratic inequalities can be solved using the Bhaskara formula, sign analysis, and graphical representation.

Identification of Coefficients a, b, and c

The coefficients a, b, and c in a quadratic inequality play fundamental roles in determining the roots and the concavity of the parabola represented by the inequality. The coefficient 'a' determines the opening and orientation of the parabola, while 'b' and 'c' influence its positioning on the Cartesian plane.

• The coefficient 'a' determines the concavity of the parabola: if a > 0, the parabola opens upwards; if a < 0, it opens downwards.

• The coefficients 'b' and 'c' influence the horizontal and vertical displacement of the parabola.

• The correct identification of the coefficients is crucial for applying the Bhaskara formula and for graphical analysis of the inequality.

Solving Quadratic Inequalities

To solve a quadratic inequality, we first solve the associated quadratic equation (ax² + bx + c = 0) using the Bhaskara formula. Then, we analyze the sign of the quadratic function in the intervals determined by the roots of the equation. From this analysis, we determine the intervals that satisfy the original inequality.

• Use the Bhaskara formula to find the roots of the associated quadratic equation (ax² + bx + c = 0).

• Analyze the sign of the quadratic function in the intervals defined by the roots.

• Determine the intervals that satisfy the condition imposed by the inequality (ax² + bx + c > 0, < 0, ≤ 0, or ≥ 0).

Practical Applications

• In civil engineering, quadratic inequalities are used to calculate the strength of structures and materials, ensuring that bridges and buildings can support specific loads.
• In finance, quadratic inequalities help model and assess the viability of investments, considering different economic scenarios.
• In information technology, quadratic inequalities are applied in optimizing algorithms, improving the performance and efficiency of computing systems.

Key Terms

• Quadratic Inequality: A mathematical expression of the type ax² + bx + c > 0 where a, b, and c are real coefficients and a ≠ 0.

• Coefficient: A multiplying number of a variable in an equation or inequality. In the case of quadratic inequalities, the coefficients are a, b, and c.

• Bhaskara Formula: A formula used to find the roots of a quadratic equation: x = (-b ± √(b² - 4ac)) / 2a.

• Sign of the Inequality: The analysis of the sign of the quadratic function in the intervals determined by the roots to determine the intervals that satisfy the inequality.

Questions

• How can the understanding and resolution of quadratic inequalities influence decisions in engineering and civil construction?

• In what ways can the techniques learned to solve quadratic inequalities be applied to optimize processes in the financial market?

• What is the importance of analyzing the sign of the quadratic function when solving quadratic inequalities in practical contexts?

Conclusion

To Reflect

Quadratic inequalities are a versatile mathematical tool with various practical applications in fields such as engineering, economics, and information technology. Understanding and solving these inequalities allow us to predict behaviors and optimize processes, which is essential in a world driven by data analysis and solving complex problems. By mastering these skills, you will be better prepared to face technical and scientific challenges, applying theoretical knowledge to real-world situations in the job market and everyday life.

Mini Challenge - Practical Challenge: Building a Safe Bridge

Use your knowledge of quadratic inequalities to solve a practical civil engineering problem.

• Form groups of 4-5 students.
• Read the problem: 'A company wants to build a bridge that can support a maximum load of 500 tons. The load on the bridge is given by the formula C(x) = 2x² + 5x - 200, where x is the number of trucks passing over the bridge. Determine the maximum number of trucks that the bridge can support.'
• Use the Bhaskara formula to solve the inequality and find the solution.
• Analyze the sign of the quadratic function to determine the intervals that satisfy the condition imposed by the inequality.
• Present your solutions and explain the reasoning used.
• Compare different approaches and solutions with the other groups.