EQUAÇÃO DO 2 GRAU 9° ANO \Prof. Gis/ aula 1

Introduction

This tutorial provides a step-by-step guide to understanding and solving second-degree equations, also known as quadratic equations. These equations are fundamental in mathematics and are often encountered in various real-world applications, such as physics, engineering, and economics. By the end of this guide, you will be able to identify the components of a quadratic equation, classify it, and solve incomplete equations.

Step 1: Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of the form:

ax² + bx + c = 0

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

Key Points

• The highest exponent of the variable (x) is 2.
• If 'a' is zero, the equation becomes linear (1st degree).

Step 2: Classifying the Quadratic Equation

Quadratic equations can be classified into two categories:

1. Complete Quadratic Equation

   • Contains all three coefficients (a, b, and c).
   • Example: 2x² + 3x + 1 = 0

2. Incomplete Quadratic Equation

   • Lacks one or both of the coefficients (b or c).
   • Examples:
     • If b = 0: 2x² + 0x + 1 = 0 (can be simplified to 2x² + 1 = 0)
     • If c = 0: 2x² + 3x + 0 = 0 (can be simplified to 2x² + 3x = 0)

Tip

Identifying the type of quadratic equation is crucial for choosing the correct solving method.

Step 3: Solving an Incomplete Quadratic Equation

To solve an incomplete quadratic equation, follow these steps:

Example Equation

Consider the incomplete equation 2x² + 3x = 0.

1. Factor the Equation:

   • Factor out the common term (x):

     x(2x + 3) = 0
     

2. Set Each Factor to Zero:

   • Solve for x:
     • First factor: x = 0
     • Second factor: 2x + 3 = 0 → 2x = -3 → x = -3/2

Practical Advice

• Always check your solutions by substituting them back into the original equation to verify they satisfy the equation.

Conclusion

In this tutorial, you learned about the structure and classification of quadratic equations, as well as the steps to solve an incomplete quadratic equation. Understanding these concepts is essential for tackling more complex mathematical problems. For further practice, refer to the exercise list provided in the video description. Continue exploring related topics to deepen your understanding of algebraic concepts.