Finding Zeros Of Quadratic Function F(x) = 2x² - 10x - 3

Introduction: Finding the Roots of a Quadratic Equation

In the realm of mathematics, quadratic functions hold a prominent position. They are characterized by their parabolic shape and are defined by the general form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. One of the fundamental aspects of understanding a quadratic function is determining its zeros, also known as roots or x-intercepts. The zeros of a quadratic function are the values of x for which the function evaluates to zero, i.e., f(x) = 0. Finding these zeros provides valuable insights into the behavior and properties of the quadratic function. In this comprehensive guide, we will delve into the process of finding the zeros of the quadratic function f(x) = 2x² - 10x - 3, employing the quadratic formula, a powerful tool for solving quadratic equations.

Understanding the zeros of a quadratic function is crucial for various applications in mathematics, physics, engineering, and other fields. For instance, in physics, the zeros of a projectile's trajectory equation represent the points where the projectile hits the ground. In engineering, the zeros of a structural equation might indicate points of instability or failure. Therefore, mastering the techniques for finding zeros is essential for problem-solving and analysis in diverse contexts. The quadratic formula, which we will explore in detail, provides a systematic approach to finding these zeros, regardless of the complexity of the equation. It is a versatile tool that can be applied to any quadratic equation, making it an indispensable part of the mathematician's toolkit. We will walk through each step of the process, ensuring a clear understanding of how to apply the quadratic formula effectively. Let's embark on this mathematical journey to uncover the zeros of our given quadratic function.

The Quadratic Formula: A Powerful Tool

The quadratic formula is a cornerstone of algebra, providing a direct method for solving quadratic equations of the form ax² + bx + c = 0. This formula elegantly expresses the solutions (zeros) in terms of the coefficients a, b, and c. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

The formula reveals that a quadratic equation can have up to two distinct real roots, one repeated real root, or two complex roots, depending on the discriminant (b² - 4ac). The discriminant plays a pivotal role in determining the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one repeated real root; and if it is negative, there are two complex roots. This information is invaluable in understanding the behavior of the quadratic function and its graph. For example, if the discriminant is positive, the parabola intersects the x-axis at two distinct points, representing the two real roots. If the discriminant is zero, the parabola touches the x-axis at one point, indicating a repeated real root. And if the discriminant is negative, the parabola does not intersect the x-axis, signifying that the roots are complex. Understanding the relationship between the discriminant and the roots is crucial for a complete analysis of quadratic equations. The quadratic formula not only provides the solutions but also offers insights into the characteristics of the solutions themselves. It is a powerful tool that simplifies the process of solving quadratic equations, making it accessible and efficient.

Applying the Quadratic Formula to f(x) = 2x² - 10x - 3

To find the zeros of the quadratic function f(x) = 2x² - 10x - 3, we will now meticulously apply the quadratic formula. First, we identify the coefficients: a = 2, b = -10, and c = -3. These coefficients are the key ingredients that we will plug into the formula to unlock the solutions. Substituting these values into the quadratic formula, we get:

x = (-(-10) ± √((-10)² - 4 * 2 * -3)) / (2 * 2)

This step involves careful substitution, ensuring that each coefficient is placed correctly within the formula. Next, we simplify the expression step by step, following the order of operations. First, we address the negative signs and the multiplication within the square root:

x = (10 ± √(100 + 24)) / 4

Now, we further simplify the expression under the square root:

x = (10 ± √124) / 4

To simplify the square root, we look for perfect square factors of 124. We find that 124 = 4 * 31, so we can rewrite the expression as:

x = (10 ± √(4 * 31)) / 4

x = (10 ± 2√31) / 4

Finally, we can simplify the expression by dividing both the numerator and the denominator by 2:

x = (5 ± √31) / 2

This gives us the two zeros of the quadratic function, which we can express separately as:

x₁ = (5 - √31) / 2

x₂ = (5 + √31) / 2

These are the precise values of x for which the function f(x) = 2x² - 10x - 3 equals zero. By meticulously applying the quadratic formula and simplifying the resulting expression, we have successfully found the zeros of the given quadratic function.

The Zeros: x = (5/2) - (√31/2) and x = (5/2) + (√31/2)

The zeros of the quadratic function f(x) = 2x² - 10x - 3, as we have meticulously calculated using the quadratic formula, are indeed x = (5/2) - (√31/2) and x = (5/2) + (√31/2). These values represent the points where the parabola defined by the quadratic function intersects the x-axis. They are the solutions to the equation 2x² - 10x - 3 = 0. The two distinct real roots indicate that the parabola crosses the x-axis at two different points, which are located at approximately x ≈ -0.298 and x ≈ 5.298. These zeros provide valuable information about the behavior of the quadratic function. For instance, they help us understand the intervals where the function is positive or negative. Between the two zeros, the function will have the opposite sign of the leading coefficient (in this case, 2), and outside the zeros, the function will have the same sign as the leading coefficient. This information is useful for sketching the graph of the parabola and understanding its overall shape and position. The zeros also play a crucial role in solving related problems, such as finding the vertex of the parabola or determining the maximum or minimum value of the function. In summary, the zeros of a quadratic function are not just solutions to an equation; they are key features that provide a deeper understanding of the function's properties and behavior.

Conclusion: Mastering Quadratic Equations

In this comprehensive exploration, we have successfully navigated the process of finding the zeros of the quadratic function f(x) = 2x² - 10x - 3. We have meticulously applied the quadratic formula, a fundamental tool in algebra, to arrive at the solutions x = (5/2) - (√31/2) and x = (5/2) + (√31/2). This journey has not only provided us with the specific zeros of this particular function but has also reinforced the broader understanding of quadratic equations and their solutions. The quadratic formula is a versatile and powerful technique that can be applied to any quadratic equation, regardless of its complexity. By mastering this formula and the steps involved in its application, you are well-equipped to tackle a wide range of mathematical problems and real-world scenarios involving quadratic relationships. Understanding the zeros of a quadratic function is crucial for various applications in fields such as physics, engineering, and economics. They represent key points in the behavior of the function, such as where it intersects the x-axis or where it reaches its maximum or minimum value. Therefore, the ability to find and interpret these zeros is an essential skill for any student of mathematics and related disciplines. As you continue your mathematical journey, remember that the quadratic formula is a valuable tool in your arsenal, ready to be deployed whenever you encounter a quadratic equation. With practice and perseverance, you will become proficient in solving these equations and unlocking the secrets they hold.