Context

Second-degree equations are a fundamental part of mathematics, used in a wide variety of fields. These equations are capable of modeling a series of phenomena in nature and in elegant problem resolutions.

To understand quadratic equations, it is necessary to comprehend the elements that compose it: coefficients (a, b, c), the roots of the equation, and the famous Δ (delta), which is the discriminant of the equation.

In our daily lives, often without realizing it, we are confronted with situations that can be described by these equations. In engineering, physics, and finance, for example, second-degree equations model the trajectory of a moving object, profit optimization, and even the determination of uncertainties.

Introduction

The project proposed aims to deeply understand quadratic equations, particularly the Bhaskara's formula (or quadratic formula), a general solution for the second-degree equation.

Bhaskara Akaria was an Indian mathematician and the author of one of the first algebra reference texts, where he proposes a solution for the quadratic equation. The formula, which became a milestone in mathematics, seeks to find the roots of the equation, and this is done through operations involving the coefficients (a, b, c) and Δ.

As a topic of fundamental importance, the understanding of second-degree equations, along with the quadratic formula, will be the basis for solving various problems within mathematics and its applications.

Practical Activity

Activity Title: Bhaskara Challenge

Project Objective:

Use the Bhaskara's formula to solve second-degree equations found in everyday applications, both theoretical and practical.

Detailed Project Description:

Students divided into groups of 3 to 5 members should identify everyday situations that can be represented by second-degree equations. The challenge will be to translate them into mathematical equations and solve them using the Bhaskara's formula.

The groups should prepare a presentation demonstrating the chosen situation, explaining how they represented it with a second-degree equation, how they arrived at the solution using the Bhaskara's formula, and the relevance of the solution found for the practical situation.

The activity should be developed in one week, with an estimated time of two to four hours of work per student.

Required Materials:

• Paper and pen for notes and calculations
• Computer with internet access for research and presentation preparation
• Software for creating slides (Powerpoint, Google Slides, etc.)
• Camera or cellphone to document the presentation

Detailed Step-by-Step for Activity Execution:

1. Group Formation: Groups will be formed by 3 to 5 students, who will work together during the project.

2. Situation Identification: Each group should identify an everyday situation that can be represented by a second-degree equation. This situation can be a problem to be solved, a decision to be made, or any other situation that can be mathematically modeled.

3. Mathematical Representation: Once the situation is identified, the group should translate it into a second-degree equation, explaining how they arrived at this representation.

4. Equation Resolution: The group should then solve the equation using the Bhaskara's formula, showing all calculations and explaining each step.

5. Presentation Preparation: The group will prepare a slide presentation documenting the entire process, from the situation identification, through the mathematical representation and equation resolution, concluding with an explanation of the relevance of the solution found for the practical situation.

6. Project Presentation: Each group will present their work to the class, where other groups will have the opportunity to ask questions and make comments.

Project Deliverables:

At the end of the project, each group must deliver the following:

1. Slide Presentation: A file containing the slide presentation with all the information and calculations performed during the project.

2. Written Report: A detailed report documenting the entire process, covering the following topics:

   • Introduction: Should contextualize the theme, its relevance and real-world application, as well as the objective of this project.
   • Development: Should explain the theory behind second-degree equations and the quadratic formula, explain the activity performed in detail, indicate the methodology used, and finally present and discuss the results obtained.
   • Conclusion: Should summarize the main points, explain the learnings obtained, and draw conclusions about the project.
   • Bibliography: Should indicate the sources they relied on to work on the project such as books, web pages, videos, etc.