Determining Real Zeros Of Quadratic Functions Using The Discriminant

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In mathematics, specifically within the realm of algebra, understanding the nature of roots for quadratic equations is crucial. A powerful tool for this is the discriminant, a part of the quadratic formula that reveals whether a quadratic equation has two real roots, one real root, or no real roots. This article will delve into how the discriminant, represented by the formula b^2 - 4ac, helps us determine the number of real zeros a quadratic function possesses. We'll explore this concept through several examples, providing a comprehensive guide for students and enthusiasts alike.

Understanding the Discriminant

The discriminant is derived from the quadratic formula, which is used to find the solutions (or roots) of a quadratic equation in the form of ax^2 + bx + c = 0. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

The expression inside the square root, b^2 - 4ac, is the discriminant. Its value dictates the nature of the roots:

  • If b^2 - 4ac > 0, the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.
  • If b^2 - 4ac = 0, the quadratic equation has exactly one real root (a repeated root). The parabola touches the x-axis at only one point.
  • If b^2 - 4ac < 0, the quadratic equation has no real roots. The roots are complex numbers, and the parabola does not intersect the x-axis.

To effectively utilize the discriminant, it’s essential to correctly identify the coefficients a, b, and c from the quadratic function. The coefficient a is the number multiplying the x^2 term, b is the coefficient of the x term, and c is the constant term. This foundational understanding is paramount in accurately determining the nature of the roots and, consequently, the zeros of the quadratic function. Mastering this concept not only aids in solving quadratic equations but also provides insights into the graphical representation of these functions, enhancing your overall comprehension of quadratic relationships.

Applying the Discriminant to Determine Real Zeros

Now, let's apply this knowledge to determine which of the given functions have two real number zeros. This involves calculating the discriminant for each function and analyzing its value. We'll consider the following functions:

  1. f(x) = x^2 + 6x + 8
  2. g(x) = x^2 + 4x + 8
  3. h(x) = x^2 - 12x + 32
  4. k(x) = x^2 + 4

For each function, we will identify the coefficients a, b, and c, substitute them into the discriminant formula (b^2 - 4ac), and then interpret the result.

Function 1: f(x) = x^2 + 6x + 8

Here, a = 1, b = 6, and c = 8. Let's calculate the discriminant:

Discriminant = b^2 - 4ac = 6^2 - 4(1)(8) = 36 - 32 = 4

Since the discriminant is 4, which is greater than 0, the function f(x) = x^2 + 6x + 8 has two distinct real zeros. This means that the graph of this quadratic function, a parabola, intersects the x-axis at two different points. These points represent the real solutions to the equation f(x) = 0, and they are critical in understanding the behavior of the quadratic function. Furthermore, the positive discriminant suggests that the function can be factored into two distinct linear factors, each corresponding to one of the real zeros. The ability to quickly determine the nature of roots using the discriminant is a fundamental skill in algebra, enabling a deeper analysis of quadratic functions and their applications.

Function 2: g(x) = x^2 + 4x + 8

In this case, a = 1, b = 4, and c = 8. The discriminant is:

Discriminant = b^2 - 4ac = 4^2 - 4(1)(8) = 16 - 32 = -16

The discriminant is -16, which is less than 0. Therefore, the function g(x) = x^2 + 4x + 8 has no real zeros. This implies that the parabola representing this function does not intersect the x-axis at any point. The roots of the equation g(x) = 0 are complex numbers, indicating a situation where the quadratic function never equals zero for any real value of x. Understanding that a negative discriminant signifies complex roots is crucial in distinguishing different types of quadratic equations and their graphical representations. This knowledge is not only useful in algebra but also in various fields where quadratic models are applied, such as physics and engineering.

Function 3: h(x) = x^2 - 12x + 32

For h(x) = x^2 - 12x + 32, we have a = 1, b = -12, and c = 32. The discriminant is calculated as follows:

Discriminant = b^2 - 4ac = (-12)^2 - 4(1)(32) = 144 - 128 = 16

Since the discriminant is 16, which is greater than 0, the function h(x) = x^2 - 12x + 32 has two distinct real zeros. This result indicates that the parabola intersects the x-axis at two different points, corresponding to the two real solutions of the quadratic equation. The positive discriminant not only confirms the existence of real roots but also provides information about the factorability of the quadratic expression. In this case, the equation can be factored into two linear terms, each representing one of the points where the parabola crosses the x-axis. This understanding is essential for solving quadratic equations and analyzing the behavior of quadratic functions.

Function 4: k(x) = x^2 + 4

Here, a = 1, b = 0 (since there is no x term), and c = 4. Calculating the discriminant:

Discriminant = b^2 - 4ac = 0^2 - 4(1)(4) = 0 - 16 = -16

The discriminant is -16, which is less than 0. Thus, the function k(x) = x^2 + 4 has no real zeros. The negative discriminant indicates that the parabola does not intersect the x-axis, meaning there are no real values of x for which k(x) = 0. The solutions to the equation are complex numbers, and this is a common characteristic of quadratic functions that open upwards and have a vertex above the x-axis. Recognizing this scenario is crucial for a comprehensive understanding of quadratic functions and their properties, enabling accurate analysis and problem-solving in various mathematical and real-world contexts.

Conclusion

In conclusion, by calculating the discriminant for each quadratic function, we can effectively determine the number of real zeros. The functions f(x) = x^2 + 6x + 8 and h(x) = x^2 - 12x + 32 have two real zeros each, as their discriminants are positive. Conversely, the functions g(x) = x^2 + 4x + 8 and k(x) = x^2 + 4 have no real zeros, indicated by their negative discriminants. Understanding the discriminant is a fundamental tool in analyzing quadratic equations and their corresponding functions, providing valuable insights into their behavior and graphical representations. This knowledge is not only essential for academic success in mathematics but also for practical applications in various fields that rely on mathematical modeling.