Calculating The Discriminant Of A Quadratic Equation An Example

In the realm of mathematics, quadratic equations hold a prominent position, serving as fundamental tools for modeling and solving a myriad of real-world problems. These equations, characterized by the general form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0, possess a unique feature known as the discriminant. The discriminant, denoted as b² - 4ac, plays a crucial role in determining the nature of the roots or solutions of a quadratic equation. It acts as a powerful indicator, revealing whether the equation has two distinct real roots, one repeated real root, or two complex roots. Understanding the discriminant is essential for gaining a deeper insight into the behavior and solutions of quadratic equations.

The discriminant, b² - 4ac, is an algebraic expression derived from the coefficients of a quadratic equation in its standard form. It serves as a critical tool for analyzing the nature of the roots of the equation without actually solving for them. The discriminant provides valuable information about the number and type of solutions a quadratic equation possesses. It helps us determine whether the equation has two distinct real roots, one repeated real root (also known as a double root), or two complex roots. This information is essential in various mathematical and real-world applications where understanding the nature of the solutions is crucial.

The discriminant is not merely an abstract mathematical concept; it has practical implications in various fields. In physics, it helps determine the trajectory of projectiles, the stability of structures, and the behavior of electrical circuits. In engineering, it aids in designing bridges, buildings, and other structures that can withstand different forces and conditions. In finance, it is used to model investment returns and assess risk. By understanding the discriminant, we can gain valuable insights into the behavior of systems and make informed decisions in a wide range of situations. This makes the discriminant a fundamental tool for problem-solving in various domains.

The discriminant, represented by the expression b² - 4ac, is a critical component in the study of quadratic equations. It acts as a powerful indicator, revealing the nature of the roots of a quadratic equation without the need for solving the equation itself. The value of the discriminant directly influences the number and type of solutions the equation possesses. To fully grasp its significance, let's delve into the mechanics of how the discriminant works and what it signifies.

The discriminant is derived directly from the quadratic formula, which provides the solutions to the quadratic equation ax² + bx + c = 0. The quadratic formula is expressed as: x = (-b ± √(b² - 4ac)) / (2a). The expression under the square root, b² - 4ac, is the discriminant. The ± sign in the quadratic formula indicates that there are potentially two solutions, but the nature of these solutions depends entirely on the value of the discriminant. If the discriminant is positive, the square root yields a real number, leading to two distinct real roots. If the discriminant is zero, the square root vanishes, resulting in one repeated real root. If the discriminant is negative, the square root yields an imaginary number, indicating two complex roots.

The discriminant provides a wealth of information about the roots of a quadratic equation. A positive discriminant (b² - 4ac > 0) signifies that the quadratic equation has two distinct real roots. This means that the parabola represented by the quadratic equation intersects the x-axis at two different points. A zero discriminant (b² - 4ac = 0) indicates that the quadratic equation has one repeated real root, also known as a double root. In this case, the parabola touches the x-axis at exactly one point, representing the single solution. A negative discriminant (b² - 4ac < 0) implies that the quadratic equation has two complex roots. This means that the parabola does not intersect the x-axis at any real point, and the solutions involve imaginary numbers.

The discriminant is not just a theoretical concept; it has practical applications in various fields. In physics, it helps determine the stability of systems, the nature of oscillations, and the trajectory of projectiles. In engineering, it aids in designing structures, circuits, and control systems. In finance, it is used in portfolio optimization and risk management. Understanding the discriminant enables us to make informed decisions and solve real-world problems involving quadratic relationships. Its role as a predictor of solution types makes it a fundamental tool in mathematics and its applications.

Now, let's apply our understanding of the discriminant to a specific quadratic equation. This will provide a practical demonstration of how to calculate the discriminant and interpret its value to determine the nature of the roots. Consider the quadratic equation 0 = -2x² + 6x + 13. Our goal is to find the value of the discriminant and use it to describe the roots of this equation.

The first step is to identify the coefficients a, b, and c in the quadratic equation. In the equation 0 = -2x² + 6x + 13, a is the coefficient of the x² term, which is -2. b is the coefficient of the x term, which is 6. c is the constant term, which is 13. Identifying these coefficients correctly is crucial, as they are the building blocks for calculating the discriminant. An error in identifying these values will lead to an incorrect discriminant and, consequently, a misinterpretation of the roots.

Next, we substitute the values of a, b, and c into the discriminant formula, b² - 4ac. In this case, we have a = -2, b = 6, and c = 13. Plugging these values into the formula, we get: Discriminant = (6)² - 4(-2)(13). Now, we perform the calculations step by step. First, square the b term: 6² = 36. Next, multiply -4 by a and c: -4(-2)(13) = 104. Finally, add these two results together: 36 + 104 = 140. Therefore, the value of the discriminant for the equation 0 = -2x² + 6x + 13 is 140.

Having calculated the discriminant, we can now interpret its value to determine the nature of the roots. The discriminant is 140, which is a positive number. According to our understanding of the discriminant, a positive discriminant indicates that the quadratic equation has two distinct real roots. This means that there are two different real numbers that, when substituted for x in the equation, will make the equation true. The parabola represented by the quadratic equation will intersect the x-axis at two distinct points, corresponding to these two real roots. This interpretation provides a comprehensive understanding of the solutions to the given quadratic equation without the need to solve for the roots explicitly.

In conclusion, the discriminant, b² - 4ac, is a pivotal concept in the study of quadratic equations. It serves as a powerful indicator of the nature of the roots of a quadratic equation, providing valuable insights without the need for solving the equation itself. The discriminant allows us to determine whether the equation has two distinct real roots, one repeated real root, or two complex roots. This understanding is crucial for various mathematical and real-world applications.

Throughout this discussion, we have explored the significance of the discriminant, its derivation from the quadratic formula, and its interpretation in determining the nature of roots. We have seen how a positive discriminant indicates two distinct real roots, a zero discriminant indicates one repeated real root, and a negative discriminant indicates two complex roots. This knowledge empowers us to analyze quadratic equations more effectively and predict their solutions.

Furthermore, we have demonstrated the practical application of the discriminant through a step-by-step example. By identifying the coefficients a, b, and c in the quadratic equation 0 = -2x² + 6x + 13, we calculated the discriminant to be 140. The positive value of the discriminant led us to conclude that the equation has two distinct real roots. This example underscores the utility of the discriminant as a problem-solving tool.

In summary, the discriminant is a fundamental concept in mathematics with broad applications. Its ability to reveal the nature of roots makes it an indispensable tool for solving quadratic equations and understanding their behavior. Whether in physics, engineering, finance, or other fields, the discriminant provides valuable insights that enable us to make informed decisions and solve real-world problems effectively. A solid grasp of the discriminant is essential for anyone seeking a deeper understanding of quadratic equations and their applications.