EQUAÇÃO DO 2 GRAU FÓRMULA DE BHÁSKARA | \Prof. Gis/ AULA 2

Introduction

This tutorial will guide you through solving a second-degree equation using Bhaskara's formula, as presented by Prof. Gis. Understanding how to work with second-degree equations is crucial in algebra, as they frequently appear in various mathematical contexts.

Step 1: Understanding the Second-Degree Equation

A second-degree equation is expressed in the standard form:

ax² + bx + c = 0

• a: Coefficient of x² (cannot be zero)
• b: Coefficient of x
• c: Constant term

Classification

• Complete Equation: All coefficients (a, b, c) are present.
• Incomplete Equation: One or more coefficients are missing (b or c can be zero).

Note: If a = 0, it is no longer a second-degree equation but a first-degree one.

Step 2: Identifying Coefficients

To solve an equation, first identify the coefficients:

• Look at the equation and assign values to a, b, and c.
• Example: For the equation 2x² + 3x - 5 = 0:
  • a = 2
  • b = 3
  • c = -5

Step 3: Applying Bhaskara's Formula

Bhaskara's formula is used to find the roots of the second-degree equation:

x = (-b ± √(b² - 4ac)) / (2a)

Steps to Use the Formula

1. Calculate the Discriminant:

   • Discriminant (D) = b² - 4ac
   • Example: For a = 2, b = 3, c = -5:
     • D = (3)² - 4 * (2) * (-5) = 9 + 40 = 49

2. Determine Roots:

   • If D > 0: Two distinct real roots
   • If D = 0: One real root (repeated)
   • If D < 0: No real roots (complex roots)

3. Substitute into Bhaskara's Formula:

   • Use the calculated D to find x:
   • Example: Continuing from the above:
     • Since D = 49 (which is greater than 0):
     • x = (-3 ± √49) / (2 * 2)
     • x = (-3 ± 7) / 4

4. Simplify the Results:

   • First root:
     • x₁ = (-3 + 7) / 4 = 4 / 4 = 1
   • Second root:
     • x₂ = (-3 - 7) / 4 = -10 / 4 = -2.5

Step 4: Solving Incomplete Equations

For incomplete equations, such as ax² + c = 0 or ax² + bx = 0, you can still apply similar methods:

• If c = 0, factor the equation or use simple algebraic manipulation.
• If b = 0, the equation simplifies to ax² + c = 0, and you can isolate x².

Example of Incomplete Equation

For the equation 2x² - 8 = 0:

1. Isolate x²: 2x² = 8
2. Divide by 2: x² = 4
3. Take the square root: x = ±2

Conclusion

You've learned how to solve second-degree equations using Bhaskara's formula and how to classify equations as complete or incomplete. Remember to identify coefficients correctly and calculate the discriminant to determine the nature of the roots. For further practice, consider solving additional equations or exploring related topics in algebra.