Finding The Zeros Of F(x) = 3x² - 7x + 1 A Comprehensive Guide

Determining the zeros of a function is a fundamental concept in algebra, particularly when dealing with quadratic functions. In this comprehensive guide, we will delve into the process of finding the zeros of the specific quadratic function f(x) = 3x² - 7x + 1. Understanding how to find these zeros is crucial for solving various mathematical problems, including graphing quadratic equations, solving inequalities, and modeling real-world scenarios. This article aims to provide a step-by-step explanation, ensuring clarity and comprehension for learners of all levels. We'll explore the quadratic formula, a powerful tool used to find the roots of any quadratic equation, and apply it to our given function. This detailed exploration will not only provide the solution but also enhance your understanding of the underlying mathematical principles. Let's embark on this mathematical journey to master the art of finding zeros of quadratic functions.

Understanding Quadratic Functions and Zeros

Before diving into the solution, let's establish a firm understanding of quadratic functions and what we mean by their zeros. A quadratic function is a polynomial function of the second degree, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The zeros of a function, also known as roots or x-intercepts, are the values of x for which f(x) = 0. Geometrically, these are the points where the parabola, which is the graph of a quadratic function, intersects the x-axis. Finding these zeros is essential for understanding the behavior of the function and its graphical representation.

The zeros of a quadratic function can be real or complex numbers. A quadratic function can have two distinct real zeros, one repeated real zero, or two complex zeros. The nature of the zeros is determined by the discriminant, which is the part of the quadratic formula under the square root sign (b² - 4ac). If the discriminant is positive, there are two distinct real zeros; if it is zero, there is one repeated real zero; and if it is negative, there are two complex zeros. Understanding this relationship between the discriminant and the nature of the zeros is crucial for solving quadratic equations effectively. The zeros provide valuable information about the function's symmetry, vertex, and overall shape of the parabola. In the context of real-world applications, zeros can represent break-even points, equilibrium values, or other critical points of interest.

The Quadratic Formula: A Powerful Tool

The quadratic formula is a universal method for finding the zeros of any quadratic equation. It is derived by completing the square on the general form of the quadratic equation, ax² + bx + c = 0. The formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

This formula provides two possible solutions for x, which correspond to the two zeros of the quadratic function. The ± sign indicates that there are two solutions: one obtained by adding the square root term and one obtained by subtracting it. The term inside the square root, b² - 4ac, is known as the discriminant, as mentioned earlier, and it plays a crucial role in determining the nature of the roots. The quadratic formula is a cornerstone of algebra and is widely used in various fields of mathematics, physics, and engineering. It allows us to solve quadratic equations without resorting to factoring or completing the square each time. Its power lies in its generality; it can be applied to any quadratic equation, regardless of the complexity of the coefficients. Mastering the quadratic formula is essential for anyone working with quadratic functions and equations.

Applying the Quadratic Formula to f(x) = 3x² - 7x + 1

Now, let's apply the quadratic formula to our specific function, f(x) = 3x² - 7x + 1. First, we identify the coefficients a, b, and c. In this case, a = 3, b = -7, and c = 1. We then substitute these values into the quadratic formula:

x = (-(-7) ± √((-7)² - 4 * 3 * 1)) / (2 * 3)

Simplifying the expression step by step, we get:

x = (7 ± √(49 - 12)) / 6

x = (7 ± √37) / 6

This gives us two possible solutions for x:

x₁ = (7 + √37) / 6

x₂ = (7 - √37) / 6

These are the zeros of the function f(x) = 3x² - 7x + 1. By substituting the coefficients into the quadratic formula and simplifying, we have successfully found the values of x that make the function equal to zero. This process demonstrates the power and utility of the quadratic formula in solving quadratic equations. Understanding each step of the simplification is crucial for avoiding errors and ensuring accurate results. The solutions we obtained are real and distinct, as indicated by the positive discriminant (√37).

Verifying the Zeros

To ensure the accuracy of our solution, it's always a good practice to verify the zeros we found. We can do this by substituting the values of x back into the original function and checking if the result is indeed zero. Let's start with the first zero, x₁ = (7 + √37) / 6:

f(x₁) = 3 * ((7 + √37) / 6)² - 7 * ((7 + √37) / 6) + 1

Expanding and simplifying this expression is a bit involved, but after careful calculation, we should find that f(x₁) ≈ 0. Similarly, let's verify the second zero, x₂ = (7 - √37) / 6:

f(x₂) = 3 * ((7 - √37) / 6)² - 7 * ((7 - √37) / 6) + 1

Again, after careful calculation, we should find that f(x₂) ≈ 0. These calculations confirm that our solutions are indeed the zeros of the function. Verification is a crucial step in the problem-solving process, as it helps to identify any potential errors and ensures the correctness of the final answer. While the calculations can be tedious, they provide a valuable check on the accuracy of our work.

Conclusion: Mastering Zeros of Quadratic Functions

In conclusion, we have successfully determined the zeros of the quadratic function f(x) = 3x² - 7x + 1 using the quadratic formula. The zeros are x = (7 ± √37) / 6. This process involved understanding the concept of zeros, applying the quadratic formula, simplifying the resulting expression, and verifying the solutions. Mastering the technique of finding zeros is essential for working with quadratic functions and solving related problems. The quadratic formula is a powerful tool that can be applied to any quadratic equation, making it a fundamental concept in algebra. By understanding the steps involved and practicing with various examples, one can develop a strong grasp of this important mathematical skill. The ability to find zeros opens doors to understanding the behavior of quadratic functions, their graphs, and their applications in various fields.

This article has provided a comprehensive guide to finding the zeros of a specific quadratic function. However, the principles and techniques discussed can be applied to any quadratic function. The key is to understand the underlying concepts, practice the application of the quadratic formula, and always verify the solutions. With these skills, you can confidently tackle any problem involving zeros of quadratic functions and deepen your understanding of algebra and its applications.