EQUAÇÃO DO 2 GRAU | ANÁLISE DO DISCRIMINANTE | \Prof. Gis/ AULA 5

Introduction

This tutorial focuses on understanding the Discriminant of a quadratic equation, as presented by Prof. Gis in her mathematics lesson. The Discriminant is a crucial concept that helps determine the nature of the roots of a quadratic equation. By the end of this tutorial, you will be able to analyze the Discriminant and understand how it influences the solutions of quadratic equations.

Step 1: Understanding the Quadratic Equation

A quadratic equation takes the standard form:

[ ax^2 + bx + c = 0 ]

Where:

• a is the coefficient of ( x^2 )
• b is the coefficient of ( x )
• c is the constant term

Step 2: Define the Discriminant

The Discriminant is calculated using the formula:

[ D = b^2 - 4ac ]

Where:

• D represents the Discriminant
• b and a are coefficients from the quadratic equation

Step 3: Analyze the Discriminant

The value of the Discriminant determines the nature of the roots of the quadratic equation:

• D > 0: There are two distinct real roots.
• D = 0: There are two equal real roots (the roots are repeated).
• D < 0: There are no real roots (the solution set is empty).

Step 4: Practical Application of the Discriminant

To practically apply this knowledge, follow these steps:

1. Identify the coefficients ( a ), ( b ), and ( c ) from your quadratic equation.
2. Substitute these values into the Discriminant formula to calculate ( D ).
3. Determine the nature of the roots based on the value of ( D ):
   • If ( D > 0 ), find the roots using the quadratic formula: [ x = \frac{-b \pm \sqrt{D}}{2a} ]
   • If ( D = 0 ), use the same formula to find the single root: [ x = \frac{-b}{2a} ]
   • If ( D < 0), note that the roots are complex and not real.

Conclusion

Understanding the Discriminant helps you analyze quadratic equations effectively. Remember to always calculate ( D ) first to determine the nature of the roots before proceeding to find the roots themselves. For further study, consider reviewing related lessons on quadratic equations and practicing with various examples to solidify your knowledge. Happy studying!