Determining Real Zeros Of Quadratic Functions Using The Discriminant

Jul 9, 2025 by ADMIN 69 views

Unveiling Real Roots A Discriminant Analysis of Quadratic Functions

In the realm of quadratic functions, identifying those that possess two distinct real number zeros is a fundamental task. The discriminant, denoted as b² - 4ac, emerges as a powerful tool in this endeavor. This article delves into the application of the discriminant to determine the nature of roots for a given set of quadratic functions. We will meticulously analyze each function, calculate its discriminant, and draw conclusions about its real roots.

Understanding the Discriminant

The discriminant, b² - 4ac, is a critical component derived from the quadratic formula, which solves for the roots of a quadratic equation in the form ax² + bx + c = 0. The discriminant's value provides invaluable insights into the nature and number of roots a quadratic equation possesses. Specifically:

If b² - 4ac > 0, the quadratic equation has two distinct real roots.
If b² - 4ac = 0, the quadratic equation has one real root (a repeated root).
If b² - 4ac < 0, the quadratic equation has no real roots (two complex roots).

This understanding of the discriminant forms the bedrock of our analysis as we explore the given quadratic functions.

Analyzing Quadratic Functions Using the Discriminant

We are presented with four quadratic functions: f(x) = x² + 6x + 8, g(x) = x² + 4x + 8, h(x) = x² - 12x + 32, and k(x) = x² + 4. Our mission is to determine which of these functions exhibit two real number zeros by calculating the discriminant for each.

1. Function f(x) = x² + 6x + 8

For the function f(x) = x² + 6x + 8, we identify the coefficients as a = 1, b = 6, and c = 8. Now, we proceed to calculate the discriminant:

b² - 4ac = 6² - 4(1)(8) = 36 - 32 = 4

Since the discriminant is 4, which is greater than 0, we conclude that the function f(x) = x² + 6x + 8 has two distinct real roots. This signifies that the graph of the function intersects the x-axis at two distinct points, corresponding to the two real zeros of the function. The positive discriminant indicates a parabola that crosses the x-axis twice, a hallmark of quadratic functions with two real roots. The roots themselves can be found using the quadratic formula, but the discriminant alone confirms their existence and nature. In this case, the fact that the discriminant is a perfect square (4 = 2²) further suggests that the roots will be rational numbers, a useful observation that can simplify the subsequent root-finding process. Understanding the relationship between the discriminant and the roots allows for a deeper comprehension of the function's behavior and graphical representation. The discriminant serves as a gateway to understanding the broader characteristics of the quadratic function, including its intercepts, vertex, and overall shape. This comprehensive understanding is crucial for applications in various fields, such as physics, engineering, and economics, where quadratic functions are frequently used to model real-world phenomena. The ability to quickly determine the nature of the roots through the discriminant is a valuable skill in these contexts, allowing for efficient problem-solving and decision-making. The analysis of f(x) serves as a clear example of how the discriminant can be effectively employed to reveal the fundamental properties of a quadratic function.

2. Function g(x) = x² + 4x + 8

Moving on to the function g(x) = x² + 4x + 8, we identify the coefficients as a = 1, b = 4, and c = 8. We calculate the discriminant as follows:

b² - 4ac = 4² - 4(1)(8) = 16 - 32 = -16

The discriminant for g(x) is -16, which is less than 0. Therefore, the function g(x) = x² + 4x + 8 has no real roots. This implies that the graph of the function does not intersect the x-axis. The negative discriminant indicates that the parabola opens upwards and is entirely above the x-axis, or opens downwards and is entirely below the x-axis, depending on the sign of the leading coefficient a. In this case, since a = 1 (positive), the parabola opens upwards and its vertex lies above the x-axis, confirming that there are no real roots. This type of quadratic function, with a negative discriminant, leads to complex roots, which involve the imaginary unit i. While complex roots are not real numbers, they are crucial in many advanced mathematical and scientific applications. Understanding that a negative discriminant signals the absence of real roots is essential for efficiently analyzing quadratic functions and avoiding unnecessary attempts to find solutions that do not exist in the real number domain. The analysis of g(x) highlights the discriminant's power in quickly determining the nature of a quadratic function's roots, saving time and effort in problem-solving. The absence of real roots has significant implications for the function's graphical representation and its applicability in modeling real-world scenarios. For example, in certain physical systems, the absence of real roots might indicate that a particular state or equilibrium is unattainable. The discriminant, therefore, is not just a mathematical tool but a gateway to understanding the broader implications of quadratic functions in various contexts. By recognizing that a negative discriminant implies no real solutions, we can effectively focus our attention on other aspects of the function, such as its vertex, axis of symmetry, and behavior as x approaches positive or negative infinity. This holistic understanding is critical for a comprehensive analysis of quadratic functions.

3. Function h(x) = x² - 12x + 32

For the function h(x) = x² - 12x + 32, we identify a = 1, b = -12, and c = 32. The discriminant is calculated as:

b² - 4ac = (-12)² - 4(1)(32) = 144 - 128 = 16

With a discriminant of 16, which is greater than 0, the function h(x) = x² - 12x + 32 has two distinct real roots. The positive discriminant affirms that the parabola intersects the x-axis at two distinct points, corresponding to the two real zeros of the function. Furthermore, since the discriminant is a perfect square (16 = 4²), the roots are rational numbers, which can be readily determined through factoring or the quadratic formula. This characteristic of the discriminant providing information about the nature of the roots (rational vs. irrational) enhances its utility in analyzing quadratic functions. The presence of two real roots suggests that the function exhibits a change in sign, crossing the x-axis from negative to positive or vice versa. The location of these roots is crucial for understanding the function's behavior and its applications in various fields. For instance, in optimization problems, the roots can represent critical points where the function attains its maximum or minimum values. The analysis of h(x) underscores the discriminant's ability to provide valuable insights into the roots of a quadratic function, extending beyond mere existence to their nature and implications for the function's behavior. The discriminant serves as a bridge connecting the algebraic representation of a quadratic function to its graphical representation and its applications in real-world contexts. By efficiently calculating and interpreting the discriminant, we can gain a comprehensive understanding of the function's properties and its potential uses. This ability is invaluable in a wide range of mathematical and scientific endeavors, where quadratic functions play a pivotal role in modeling and solving problems.

4. Function k(x) = x² + 4

Lastly, for the function k(x) = x² + 4, we have a = 1, b = 0, and c = 4. The discriminant is:

b² - 4ac = 0² - 4(1)(4) = 0 - 16 = -16

The discriminant for k(x) is -16, which is less than 0. Thus, the function k(x) = x² + 4 has no real roots. The negative discriminant indicates that the parabola does not intersect the x-axis. In this case, since a = 1 (positive), the parabola opens upwards and its vertex lies above the x-axis, confirming the absence of real roots. This is a clear example of a quadratic function that has complex roots but no real roots. The function k(x) is a simple illustration of how a quadratic can be vertically shifted away from the x-axis, preventing any real number solutions. The absence of real roots has implications for the function's graphical representation and its interpretation in real-world contexts. For example, if k(x) represented the height of an object above the ground, the negative discriminant would indicate that the object never reaches the ground. This understanding is crucial for accurately interpreting mathematical models and avoiding misinterpretations. The discriminant serves as a powerful indicator of the feasibility and realism of solutions in various applications. By recognizing that a negative discriminant implies no real solutions, we can effectively assess the validity of our models and refine them if necessary. The analysis of k(x) provides a concise demonstration of the discriminant's role in determining the existence of real roots and its broader implications for understanding quadratic functions and their applications. The discriminant's ability to quickly identify the nature of roots is an invaluable tool for mathematicians, scientists, and engineers, allowing for efficient problem-solving and informed decision-making.

Conclusion

In conclusion, by calculating the discriminant for each quadratic function, we have determined that f(x) = x² + 6x + 8 and h(x) = x² - 12x + 32 have two real number zeros. This analysis underscores the power of the discriminant as a tool for understanding the nature of roots in quadratic equations, providing a foundation for further exploration and application of quadratic functions in diverse fields.