Finding Zeros Of The Quadratic Function F(x) = 6x² - 24x + 1

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Finding the zeros, also known as roots or x-intercepts, of a quadratic function is a fundamental concept in mathematics. These zeros represent the points where the parabola, the graphical representation of the quadratic function, intersects the x-axis. In this in-depth guide, we will embark on a journey to determine the zeros of the quadratic function f(x) = 6x² - 24x + 1. We will explore various methods, including the quadratic formula and completing the square, to unravel the solutions and gain a deeper understanding of the underlying principles. Understanding zeros is crucial in many areas of mathematics and its applications, including physics, engineering, and economics. Mastering the techniques for finding zeros is therefore an essential skill for anyone pursuing studies or careers in these fields. Zeros provide key information about the behavior of quadratic functions and their graphs, allowing for the solution of real-world problems and the optimization of various processes. Furthermore, the ability to manipulate and solve quadratic equations is a stepping stone to more advanced mathematical concepts and techniques.

Decoding Quadratic Functions: A Foundation for Finding Zeros

Before we dive into the methods for finding zeros, let's first establish a solid understanding of quadratic functions. A quadratic function is a polynomial function of degree two, which can be expressed in the general form:

f(x) = ax² + bx + c

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens upwards if 'a' is positive and downwards if 'a' is negative. The zeros of the quadratic function correspond to the x-coordinates of the points where the parabola intersects the x-axis. These points are also known as the roots of the quadratic equation:

ax² + bx + c = 0

Understanding the relationship between the quadratic function, its graph, and its zeros is crucial for solving various mathematical problems. The zeros not only provide information about the x-intercepts but also about the axis of symmetry and the vertex of the parabola. The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves, and the vertex is the point where the parabola changes direction. The vertex represents the minimum or maximum value of the quadratic function, depending on whether the parabola opens upwards or downwards. The zeros, the axis of symmetry, and the vertex collectively provide a comprehensive understanding of the behavior and characteristics of the quadratic function. Moreover, the coefficients 'a', 'b', and 'c' in the quadratic equation play significant roles in determining the shape and position of the parabola. The coefficient 'a' affects the steepness and direction of the parabola, while 'b' and 'c' influence its horizontal and vertical position. By analyzing these coefficients, we can predict the number and nature of the zeros, whether they are real or complex, distinct or repeated.

Method 1: Unveiling Zeros with the Quadratic Formula

The quadratic formula is a powerful tool for finding the zeros of any quadratic function. It provides a direct solution, regardless of the complexity of the equation. The formula is derived from the process of completing the square and is given by:

x = (-b ± √(b² - 4ac)) / 2a

where 'a', 'b', and 'c' are the coefficients of the quadratic function f(x) = ax² + bx + c. The ± symbol indicates that there are two possible solutions, one with the plus sign and one with the minus sign. The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant provides valuable information about the nature of the zeros:

  • If b² - 4ac > 0, the quadratic function has two distinct real zeros.
  • If b² - 4ac = 0, the quadratic function has one real zero (a repeated root).
  • If b² - 4ac < 0, the quadratic function has two complex zeros.

Applying the quadratic formula involves substituting the values of 'a', 'b', and 'c' into the formula and simplifying the expression. It's a straightforward process, but it's crucial to be meticulous with the calculations to avoid errors. The formula provides a reliable method for finding zeros, especially when factoring the quadratic expression is not feasible or straightforward. The quadratic formula is not only a mathematical tool but also a testament to the power of algebraic manipulation. It encapsulates the essence of solving quadratic equations and provides a universal solution applicable to all quadratic functions. The discriminant, a key component of the quadratic formula, serves as a diagnostic tool, revealing the nature of the zeros without explicitly calculating them. This insight allows us to anticipate the characteristics of the solutions and choose the most appropriate method for solving the equation. In addition, the quadratic formula has historical significance, representing a culmination of mathematical efforts to solve equations of degree two. Its discovery and refinement have paved the way for the development of more advanced algebraic techniques and have contributed to the progress of mathematics as a whole.

Applying the Quadratic Formula to f(x) = 6x² - 24x + 1

Now, let's put the quadratic formula into action and find the zeros of our given function, f(x) = 6x² - 24x + 1. First, we identify the coefficients:

  • a = 6
  • b = -24
  • c = 1

Next, we substitute these values into the quadratic formula:

x = (-(-24) ± √((-24)² - 4 * 6 * 1)) / (2 * 6)

Simplifying the expression, we get:

x = (24 ± √(576 - 24)) / 12
x = (24 ± √552) / 12

We can further simplify the square root by factoring out the largest perfect square:

x = (24 ± √(4 * 138)) / 12
x = (24 ± 2√138) / 12

Now, we can divide both the numerator and the denominator by 2:

x = (12 ± √138) / 6

Finally, we can separate the two solutions:

x = 2 ± √138 / 6

Therefore, the zeros of the quadratic function f(x) = 6x² - 24x + 1 are:

x = 2 + √(23/6) or x = 2 - √(23/6)

This detailed step-by-step application of the quadratic formula highlights its power and efficiency in solving quadratic equations. Each step involves careful manipulation of the algebraic expressions, ensuring accuracy and clarity. The simplification of the square root and the subsequent division by a common factor demonstrate the importance of reducing the solution to its simplest form. The final result provides two distinct real zeros, reflecting the fact that the discriminant is positive. This example serves as a practical illustration of how the quadratic formula can be used to find the zeros of any quadratic function, regardless of its complexity. The process not only yields the numerical values of the zeros but also reinforces the understanding of algebraic principles and techniques. Furthermore, this example showcases the elegance and beauty of mathematics, where a seemingly complex problem can be solved through the application of a well-established formula and a series of logical steps.

Method 2: Completing the Square – An Alternative Approach

Completing the square is another powerful method for finding the zeros of a quadratic function. This method involves transforming the quadratic expression into a perfect square trinomial, which can then be easily solved. While the quadratic formula provides a direct solution, completing the square offers a deeper understanding of the structure of quadratic equations and the relationship between their coefficients and roots. This method is particularly useful when dealing with quadratic functions where the leading coefficient is not 1, as it allows for a systematic manipulation of the equation to isolate the variable. The process of completing the square involves several key steps, including dividing the entire equation by the leading coefficient, adding and subtracting a constant term to create a perfect square trinomial, and then simplifying the equation to solve for the variable. Each step requires careful attention to detail and a solid understanding of algebraic principles. However, the rewards of mastering this method are significant, as it provides a valuable tool for solving quadratic equations and a deeper appreciation for the underlying mathematical concepts.

Steps for Completing the Square:

  1. Divide by the leading coefficient: If the coefficient of the x² term (a) is not 1, divide the entire equation by 'a'. This step ensures that the coefficient of x² is 1, which is crucial for completing the square.
  2. Isolate the constant term: Move the constant term (c) to the right side of the equation. This step prepares the equation for the addition of a constant term that will complete the square.
  3. Complete the square: Take half of the coefficient of the x term (b/2), square it ((b/2)²), and add it to both sides of the equation. This is the core step of the method, as it creates a perfect square trinomial on the left side of the equation.
  4. Factor the perfect square trinomial: The left side of the equation should now be a perfect square trinomial, which can be factored as (x + b/2)². This step simplifies the equation and allows for the isolation of the variable.
  5. Solve for x: Take the square root of both sides of the equation and solve for x. Remember to consider both the positive and negative square roots. This step yields the two solutions for the quadratic equation.

Completing the square is not just a method for solving quadratic equations; it is also a fundamental technique in various areas of mathematics, including calculus and analytic geometry. It provides a powerful tool for transforming quadratic expressions into a more manageable form, allowing for the solution of a wide range of problems. The process of completing the square also sheds light on the relationship between the coefficients of a quadratic equation and the position and shape of its graph, the parabola. By completing the square, we can rewrite the quadratic function in vertex form, which directly reveals the coordinates of the vertex and the axis of symmetry. This connection between algebra and geometry makes completing the square a valuable tool for both theoretical and applied mathematics.

Applying Completing the Square to f(x) = 6x² - 24x + 1

Let's apply the method of completing the square to our quadratic function, f(x) = 6x² - 24x + 1. We start by setting the function equal to zero:

6x² - 24x + 1 = 0

  1. Divide by the leading coefficient: Divide both sides of the equation by 6:

x² - 4x + 1/6 = 0 ```

  1. Isolate the constant term: Move the constant term to the right side:

x² - 4x = -1/6 ```

  1. Complete the square: Take half of the coefficient of the x term (-4/2 = -2), square it ((-2)² = 4), and add it to both sides:

x² - 4x + 4 = -1/6 + 4 ```

  1. Factor the perfect square trinomial: Factor the left side as a perfect square:

(x - 2)² = 23/6 ```

  1. Solve for x: Take the square root of both sides:

x - 2 = ±√(23/6) ```

Add 2 to both sides:

```

x = 2 ± √(23/6) ```

Thus, the zeros of the quadratic function f(x) = 6x² - 24x + 1 are:

x = 2 + √(23/6) or x = 2 - √(23/6)

This detailed walkthrough demonstrates the step-by-step application of completing the square. Each step builds upon the previous one, leading to the solution in a logical and systematic manner. The process involves algebraic manipulation, including division, addition, and factoring, showcasing the interconnectedness of these concepts. The final result, the zeros of the quadratic function, is consistent with the result obtained using the quadratic formula, validating the effectiveness of both methods. Completing the square not only provides a solution to the quadratic equation but also offers insights into the structure of the equation and the relationship between its coefficients and roots. This method is a valuable tool for both solving quadratic equations and deepening the understanding of quadratic functions.

Conclusion: Mastering Zeros of Quadratic Functions

In this comprehensive guide, we have explored the concept of zeros of quadratic functions and delved into two powerful methods for finding them: the quadratic formula and completing the square. We applied these methods to the specific function f(x) = 6x² - 24x + 1 and successfully determined its zeros. Mastering these techniques is crucial for a strong foundation in mathematics, as quadratic functions play a vital role in various applications. The zeros of a quadratic function provide valuable information about its behavior, including its intercepts, turning points, and overall shape. Understanding these characteristics is essential for solving real-world problems and making informed decisions in fields such as physics, engineering, and economics. The quadratic formula and completing the square are not just mathematical tools; they are also gateways to deeper mathematical understanding. By mastering these methods, students develop a stronger sense of algebraic manipulation, problem-solving strategies, and the interconnectedness of mathematical concepts. Furthermore, the ability to solve quadratic equations is a prerequisite for tackling more advanced topics, such as calculus and differential equations. Therefore, investing time and effort in mastering these techniques is an investment in future mathematical success. In addition to the practical applications of quadratic functions, the study of their zeros also provides a glimpse into the beauty and elegance of mathematics. The systematic approach to solving quadratic equations, whether through the quadratic formula or completing the square, showcases the power of logical reasoning and algebraic manipulation. The consistent results obtained from different methods reinforce the reliability and validity of mathematical principles. The exploration of quadratic functions and their zeros is a journey that not only enhances mathematical skills but also cultivates a deeper appreciation for the subject as a whole.

Therefore, the correct answer is:

A. x = 2 + √(23/6) or x = 2 - √(23/6)