Introduction-to-Quadratic-Equations.pptx

Mathematics 9 Quarter 1 week 4

1. Introduction to Quadratic Equations Quadratic equations are polynomial equations with a highest degree of 2. They are characterized by their parabolic graphs and play a significant role in various fields, including physics, engineering, and economics. by EMILY BAUTISTA

2. The Standard Form of a Quadratic Equation The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The coefficients a, b, and c determine the shape and position of the parabola that represents the equation. Coefficient a Determines the direction of the parabola. Coefficient b Influences the slope and position of the parabola. Coefficient c Represents the y-intercept of the parabola.

3. The Discriminant of a Quadratic Equation The discriminant is a part of the quadratic formula that helps determine the nature of the roots of a quadratic equation. It is represented by the expression b² - 4ac. 1 Nature of Roots The discriminant reveals whether the quadratic equation has real or complex roots, and whether these roots are distinct or repeated. 2 Number of Solutions The discriminant also indicates the number of solutions that the equation possesses, which can be two, one, or zero.

4. Calculating the Discriminant To calculate the discriminant, simply substitute the values of a, b, and c from the quadratic equation into the formula b² - 4ac. The result will be a numerical value that can be interpreted to understand the nature of the roots. Equation: ax² + bx + c = 0 Discriminant: b² - 4ac

5. Interpreting the Discriminant The discriminant provides valuable information about the roots of a quadratic equation. By analyzing the sign of the discriminant, you can determine the nature and number of solutions. Positive Discriminant The equation has two distinct real roots. The parabola intersects the x-axis at two distinct points. Negative Discriminant The equation has two complex roots. The parabola does not intersect the x-axis. Zero Discriminant The equation has one repeated real root. The parabola touches the x-axis at one point.

6. Positive, Negative, and Zero Discriminants Depending on the sign of the discriminant, the quadratic equation can have different types of roots. Understanding the relationship between the discriminant and the roots is crucial for solving quadratic equations. Positive Discriminant Two distinct real roots Zero Discriminant One repeated real root Negative Discriminant Two complex roots

7. Applications of the Discriminant The discriminant has various applications in diverse fields. It can be used to determine the feasibility of a solution, to analyze the stability of a system, and to predict the behavior of physical phenomena. Engineering Analyzing the stability of structures and systems. Physics Predicting the trajectory of projectiles and the motion of objects. Economics Determining the equilibrium points in economic models. Finance Analyzing the profitability of investments.

8. Conclusion The discriminant is a powerful tool for understanding the nature of roots of quadratic equations. It provides valuable information that can be applied in various fields, making it an essential concept in mathematics.