RAÍZES DE UMA EQUÇÃO DO 2 GRAU | \Prof. Gis/ AULA 4

Introduction

This tutorial provides a comprehensive guide on understanding the roots of a quadratic equation, specifically tailored for students looking to grasp the concepts introduced in the video by Prof. Gis. Quadratic equations are fundamental in algebra, and this guide will simplify their structure and classification for easier learning.

Step 1: Understand the Structure of a Quadratic Equation

A quadratic equation is defined as a polynomial equation of degree two. It is typically expressed in the standard form:

ax^2 + bx + c = 0

• Coefficients:

  • a: Coefficient of x² (must not be zero).
  • b: Coefficient of x (can be zero).
  • c: Constant term (can be zero).

• Importance of the Coefficient a:

  • If a = 0, the equation becomes linear, not quadratic.

Step 2: Classify the Quadratic Equation

Quadratic equations can be classified into two categories:

1. Complete Quadratic Equation:

   • Contains all three coefficients (a, b, and c).
   • Example: 2x^2 + 3x + 1 = 0

2. Incomplete Quadratic Equation:

   • Lacks at least one of the coefficients (b or c).
   • Examples:
     • If b = 0: 3x^2 + 4 = 0
     • If c = 0: x^2 - 5x = 0

Step 3: Identifying the Roots of the Quadratic Equation

The roots of a quadratic equation are the values of x that satisfy the equation. These can be found using the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

• Discriminant (D = b² - 4ac):
  • D > 0: Two distinct real roots.
  • D = 0: One real root (repeated).
  • D < 0: No real roots (complex roots).

Step 4: Solving Quadratic Equations

To find the roots:

1. Identify coefficients a, b, and c from the equation.
2. Calculate the discriminant using D = b² - 4ac.
3. Use the quadratic formula to calculate the roots based on the value of D.

Practical Example:

Given the equation 2x^2 + 3x - 5 = 0:

1. Identify coefficients: a = 2, b = 3, c = -5.
2. Calculate the discriminant:

   D = 3² - 4(2)(-5) = 9 + 40 = 49
   

3. Calculate the roots:

   x = (-3 ± √49) / (2 * 2)
   x = (-3 ± 7) / 4
   

   • Roots: x = 1 and x = -2.5.

Conclusion

Understanding the structure and classification of quadratic equations is essential for solving them. By identifying the coefficients and using the quadratic formula, you can easily find the roots of any quadratic equation. For further practice, consider solving different quadratic equations using the steps outlined above. Happy studying!