Finding The X-Intercept Of F(x) = X^2 - 25 A Step-by-Step Guide

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In this article, we will delve into the process of identifying the x-intercept(s) of a given function. Specifically, we will focus on the function f(x) = x^2 - 25. Understanding x-intercepts is crucial in various mathematical contexts, including graphing functions, solving equations, and analyzing real-world scenarios modeled by mathematical functions. Before we dive into the solution, let's define what an x-intercept is and why it holds significance in the realm of mathematics.

Understanding X-Intercepts

The x-intercept of a function is the point where the graph of the function intersects the x-axis. At this point, the y-coordinate (or the function value, f(x)) is equal to zero. In simpler terms, it's the value(s) of x for which f(x) = 0. X-intercepts are also known as roots or zeros of the function. Finding the x-intercepts is a fundamental step in analyzing the behavior of a function and its graphical representation.

The significance of x-intercepts stems from their ability to provide key insights into the function's behavior. They indicate where the function's output transitions from positive to negative or vice versa. In real-world applications, x-intercepts can represent crucial points, such as break-even points in business models or the time when a projectile hits the ground in physics. Thus, understanding how to find x-intercepts is an essential skill in mathematics and its applications.

To find the x-intercept(s) of a function, we set the function equal to zero and solve for x. This process involves algebraic manipulation and the application of various techniques depending on the complexity of the function. For quadratic functions like the one we are considering (f(x) = x^2 - 25), factoring, using the quadratic formula, or completing the square are common methods.

In the following sections, we will apply these principles to determine the x-intercept(s) of the function f(x) = x^2 - 25. We will explore the factoring method in detail, demonstrating how to break down the quadratic expression and find the values of x that satisfy the equation f(x) = 0. This step-by-step approach will not only provide the answer to the given question but also enhance your understanding of how to find x-intercepts for similar functions.

Solving for the X-Intercepts of f(x) = x^2 - 25

To determine the x-intercepts of the function f(x) = x^2 - 25, we need to find the values of x for which f(x) = 0. This means solving the equation x^2 - 25 = 0. The given function is a quadratic function, and we can solve it using several methods, such as factoring, the quadratic formula, or completing the square. In this case, factoring is the most straightforward approach.

The expression x^2 - 25 is a difference of squares, which can be factored into the form (a - b)(a + b), where a is x and b is 5. Therefore, we can rewrite the equation as:

(x - 5)(x + 5) = 0

Now, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate equations to solve:

  1. x - 5 = 0
  2. x + 5 = 0

Solving the first equation, x - 5 = 0, we add 5 to both sides to isolate x:

x = 5

Solving the second equation, x + 5 = 0, we subtract 5 from both sides to isolate x:

x = -5

Thus, the x-intercepts of the function f(x) = x^2 - 25 are x = 5 and x = -5. These are the points where the graph of the function intersects the x-axis. To visualize this, we can imagine a parabola that opens upwards (since the coefficient of x^2 is positive) and crosses the x-axis at -5 and 5.

Now, let's relate these findings back to the original question. The question asks which of the following is an x-intercept of the function f(x) = x^2 - 25. The options provided are A. -20, B. -15, C. -25, and D. -5. Based on our calculations, the correct answer is D. -5, as it is one of the values we found for the x-intercepts.

In the next section, we will discuss why the other options are incorrect and further solidify our understanding of x-intercepts in the context of quadratic functions. This will provide a comprehensive understanding of how to identify and calculate x-intercepts, ensuring you are well-prepared to tackle similar problems.

Analyzing the Incorrect Options

Having determined that the x-intercepts of the function f(x) = x^2 - 25 are 5 and -5, it is important to understand why the other options provided (A. -20, B. -15, and C. -25) are incorrect. This analysis will not only reinforce the concept of x-intercepts but also highlight common mistakes students might make when solving similar problems. By understanding these pitfalls, you can enhance your problem-solving skills and avoid errors in future calculations.

Let's examine each incorrect option in detail:

  • A. -20: To check if -20 is an x-intercept, we would substitute x = -20 into the function f(x) = x^2 - 25 and see if the result is zero:

    f(-20) = (-20)^2 - 25 = 400 - 25 = 375

    Since f(-20) is not equal to zero, -20 is not an x-intercept of the function. This result indicates that the point (-20, 375) lies on the graph of the function, but it is not on the x-axis.

  • B. -15: Similarly, we substitute x = -15 into the function:

    f(-15) = (-15)^2 - 25 = 225 - 25 = 200

    Again, f(-15) is not equal to zero, so -15 is not an x-intercept. The point (-15, 200) is on the graph, but not on the x-axis.

  • C. -25: Substituting x = -25 into the function gives:

    f(-25) = (-25)^2 - 25 = 625 - 25 = 600

    As with the previous options, f(-25) is not zero, indicating that -25 is not an x-intercept. The point (-25, 600) is on the graph but not on the x-axis.

These calculations demonstrate that none of the incorrect options satisfy the condition for being an x-intercept, which is f(x) = 0. This exercise underscores the importance of substituting the potential x-intercept into the function and verifying that the result is indeed zero. A common mistake is to confuse x-intercepts with other features of the function, such as the y-intercept (where the graph intersects the y-axis) or points that simply lie on the graph but not on the x-axis.

In the final section, we will summarize our findings and discuss the broader implications of understanding x-intercepts in the context of quadratic functions and their applications. This will provide a comprehensive overview of the topic and reinforce the key concepts learned.

Conclusion: The Significance of X-Intercepts

In summary, we have successfully determined the x-intercepts of the function f(x) = x^2 - 25 by setting the function equal to zero and solving for x. We found that the x-intercepts are x = 5 and x = -5. Among the given options, D. -5 is the correct answer. We also analyzed why the other options (A. -20, B. -15, and C. -25) are incorrect by demonstrating that substituting these values into the function does not result in zero.

This exercise highlights the fundamental concept of x-intercepts and their importance in understanding the behavior of functions. X-intercepts, also known as roots or zeros, are the points where the graph of a function intersects the x-axis. They provide valuable information about where the function's output changes sign and are crucial in various mathematical and real-world applications.

For quadratic functions like f(x) = x^2 - 25, finding the x-intercepts often involves factoring, using the quadratic formula, or completing the square. In this case, factoring the difference of squares was the most efficient method. However, understanding different solution techniques is essential for tackling a variety of quadratic equations.

The ability to identify and calculate x-intercepts is not just a mathematical skill; it has practical implications in various fields. For example, in physics, x-intercepts can represent the time when a projectile hits the ground. In economics, they can indicate break-even points where costs equal revenue. In engineering, they can help determine the stability of systems.

Therefore, mastering the concept of x-intercepts and the techniques for finding them is a valuable investment in your mathematical toolkit. By understanding how to solve problems like the one presented here, you will be better equipped to tackle more complex mathematical challenges and apply these skills in real-world contexts. Remember to always verify your solutions and understand why incorrect options do not satisfy the conditions for being x-intercepts. This thorough approach will enhance your problem-solving abilities and lead to greater success in mathematics and beyond.