EQUAÇÃO DO 2 GRAU \Prof. Gis/ AULA 3

Introduction

This tutorial will guide you through solving second-degree equations (equações do 2º grau) using the well-known Bhaskara formula. You'll learn how to identify coefficients, classify equations as complete or incomplete, and solve examples step-by-step. Understanding these concepts is crucial for mastering quadratic equations, which are common in various mathematical applications.

Step 1: Understand the Structure of a Second-Degree Equation

• A second-degree equation is typically expressed in the form:

  ax² + bx + c = 0
  

• Here:
  • a is the coefficient of x² (it cannot be zero).
  • b is the coefficient of x.
  • c is the constant term.

Step 2: Classify the Equation

• Determine if the equation is:
  • Complete: If all coefficients (a, b, and c) are present.
  • Incomplete: If either b or c is missing.
    • Example of incomplete equations:
      • If b = 0: ax² + c = 0
      • If c = 0: ax² + bx = 0
• Remember, if a = 0, the equation becomes a first-degree equation.

Step 3: Identify the Coefficients

• Extract the values of a, b, and c from the equation.
• Example: For the equation 2x² - 4x + 1 = 0, the coefficients are:
  • a = 2
  • b = -4
  • c = 1

Step 4: Solve the Equation Using the Bhaskara Formula

• The Bhaskara formula is:

  x = (-b ± √(b² - 4ac)) / (2a)
  

• Follow these steps:
  1. Calculate the discriminant (Δ):

     Δ = b² - 4ac
     

  2. Determine the nature of the roots:
     • If Δ > 0, there are two distinct real roots.
     • If Δ = 0, there is one real root (repeated).
     • If Δ < 0, there are no real roots.
  3. Substitute values into the Bhaskara formula to find x.

Step 5: Example of Solving a Complete Equation

• Consider the equation 2x² - 4x + 1 = 0.
  1. Identify coefficients:
     • a = 2, b = -4, c = 1
  2. Calculate the discriminant:

     Δ = (-4)² - 4 * 2 * 1 = 16 - 8 = 8
     

  3. Since Δ > 0, there are two distinct real roots.
  4. Use the Bhaskara formula:

     x1 = (4 + √8) / (2 * 2)
     x2 = (4 - √8) / (2 * 2)
     

Conclusion

You have learned how to identify and classify second-degree equations, extract coefficients, and solve them using the Bhaskara formula. Practice with various examples to solidify your understanding. For further study, consider exploring related lessons on first-degree equations and algebraic expressions. Happy studying!