Contextualization

The Quadratic Equation is one of the fundamental bases of mathematics, serving as a starting point for the study of functions and for solving various problems involving movements, electrical and mechanical systems, economics, engineering, physics, among several other fields. It was around the 9th century that the Persian mathematician Al-Khwarizmi introduced methods for solving linear and quadratic equations, contributing significantly to the development of algebra. The 'Bhaskara's formula', as it is commonly known, was named in honor of the great Indian mathematician Bhaskara Akaria, who conducted important studies in the area. The formula allows finding the roots of a second-degree equation in a systematic and simple way.

The Quadratic Equation is an equation of the form ax² + bx + c = 0, where x represents a variable, and a, b, and c are constants. The term 'quadratic' comes from the word 'square', since x is squared. In this context, the 'solution' or 'roots' of the quadratic equation are the values that x can assume, so that the equation is satisfied (that is, the equation equals zero). Bhaskara's formula is an efficient way to find these roots and is an indispensable tool for any math student.

Importance of the Theme

Solving second-degree equations is a fundamental skill in mathematics. Furthermore, this concept has enormous applicability in various other disciplines, such as physics, chemistry, engineering, economics, among others. In the real world, second-degree equations are everywhere: in the trajectory of a projectile, in determining the time to pay off a loan, in optimizing production processes, in calculating areas, in determining distances and time in movements, etc.

Moreover, learning to solve second-degree equations develops logical and critical thinking skills, which are useful not only in the academic field but also in everyday life. Therefore, in addition to its intrinsic value as a fundamental topic in the field of mathematics, the study of the second-degree equation is also valuable for its practical applications and for the impact it has on the development of students' intellectual skills.

Atividade Prática

Activity Title: 'Unraveling the Quadrature of the Circle with Bhaskara'

Project Objective

The main objective of this project is to align theory and practice in solving quadratic equations using Bhaskara's formula, deepening students' understanding of this important mathematical concept. Additionally, the project aims to develop competencies such as teamwork, communication, time management, and creative thinking.

Project Description

In this project, students are challenged to create their own second-degree equations and demonstrate their resolution using Bhaskara's formula. The proposed scenario is the modeling of the area of a circle through quadratic equations. The activity will be carried out in groups of 3 to 5 students and must be completed within one month.

Required Materials

Students will need the following materials:

• Paper and pencil for notes and calculations
• Compass to draw circles
• Ruler to measure distances
• Calculator to assist in calculations
• Computer with internet access for research and report writing

Project Step by Step

1. Form groups of 3 to 5 students. Each group must elect a leader to coordinate activities and ensure that all members participate and contribute to the project.

2. Each group will have to create three different circles with distinct radii and, based on them, develop three different quadratic equations that represent the area of the circles as a function of the radius.

3. Solve the created second-degree equations using Bhaskara's formula.

4. Analyze the results obtained. Does the value of x (circle radius) obtained through the equation resolution coincide with the radius of the circle used in formulating the equation? Discuss the results and draw conclusions.

5. Write a detailed report on the project, including introduction, development, conclusion, and bibliography. The report should follow the outline below:

   • Introduction: Should contextualize the theme of the Quadratic Equation and Bhaskara's formula, explain the fundamental concepts, their relevance, practical applications, and the project's objective.

   • Development: Should detail the methodology used, explain the creation of quadratic equations and their resolution with Bhaskara's formula, and present the results obtained. Visual elements (such as graphs or images) that help illustrate the work done should be included.

   • Conclusion: Should summarize the main points of the project, highlight the learnings obtained, the skills developed, and the conclusions drawn from the results.

   • Bibliography: Should cite all sources of information used during the project, whether books, videos, websites, among others.

Project Deliverables

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

1. The second-degree equations created from the circles, as well as their resolutions with Bhaskara's formula

2. The detailed report following the proposed outline above.