Finding The Roots Of X² - 48 A Step-by-Step Solution
In mathematics, determining the roots of a function is a fundamental skill, especially when dealing with quadratic equations. Quadratic equations, which are polynomial equations of the second degree, often appear in various mathematical contexts and real-world applications. Understanding how to find their roots is essential for solving problems in physics, engineering, economics, and many other fields. The roots of a function, also known as the zeros or solutions, are the values of the variable (usually denoted as 'x') that make the function equal to zero. In other words, they are the points where the graph of the function intersects the x-axis. For a quadratic function in the form of f(x) = ax² + bx + c, finding these roots involves techniques such as factoring, completing the square, or using the quadratic formula. Each method offers a unique approach to solving the equation, and the choice of method often depends on the specific characteristics of the equation. For example, factoring is most effective when the quadratic expression can be easily factored into two binomials. The quadratic formula, on the other hand, provides a universal solution that works for any quadratic equation, regardless of its factorability. Mastering these techniques not only enhances problem-solving skills but also provides a deeper understanding of the behavior of quadratic functions and their applications in various disciplines. This article will delve into a detailed exploration of how to find the roots of the quadratic function f(x) = x² - 48, illustrating the process step by step and highlighting the underlying mathematical principles.
Understanding Quadratic Equations
Before diving into the specific problem, let's briefly review the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The condition that 'a' cannot be zero is crucial because if 'a' were zero, the equation would reduce to a linear equation (bx + c = 0), which is a different type of equation with a different solution method. The coefficients 'a', 'b', and 'c' play significant roles in determining the shape and position of the parabola, which is the graphical representation of a quadratic function. The coefficient 'a' dictates the direction in which the parabola opens: if 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards. The magnitude of 'a' also affects the 'width' of the parabola; a larger absolute value of 'a' results in a narrower parabola, while a smaller absolute value results in a wider parabola. The coefficient 'b' influences the horizontal position of the parabola's vertex, and the constant term 'c' determines the y-intercept of the parabola. Understanding these properties is essential for sketching the graph of a quadratic function and for predicting the behavior of the function. The roots of a quadratic equation are the x-values for which f(x) = 0. These roots are also the x-intercepts of the parabola. A quadratic equation can have two distinct real roots, one repeated real root, or two complex roots, depending on the discriminant (b² - 4ac). The discriminant provides valuable information about the nature of the roots without actually solving the equation. 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. Understanding these fundamental concepts is key to mastering the techniques for solving quadratic equations and applying them effectively in various mathematical and real-world contexts.
The Given Function: f(x) = x² - 48
Our specific function is f(x) = x² - 48. This is a quadratic function where a = 1, b = 0, and c = -48. Notice that the 'b' term is zero, which simplifies the process of finding the roots. The absence of the 'bx' term indicates that the parabola is symmetric about the y-axis, and its vertex lies on the y-axis. This symmetry is a key characteristic that simplifies the analysis and solution of the equation. The coefficient 'a' being 1 means the parabola opens upwards, and its shape is relatively standard compared to parabolas with larger or smaller 'a' values. The constant term 'c' being -48 tells us that the parabola intersects the y-axis at the point (0, -48). This y-intercept provides a crucial reference point when sketching the graph of the function. To find the roots, we need to solve the equation x² - 48 = 0. This means we are looking for the x-values that make the function equal to zero, which correspond to the points where the parabola intersects the x-axis. Solving this equation will give us the x-coordinates of these intersection points, which are the roots of the function. There are several methods we can use to solve this equation, including factoring, using the square root property, or applying the quadratic formula. In this case, the absence of the 'bx' term makes the square root property the most straightforward and efficient method. By isolating the x² term and then taking the square root of both sides, we can quickly determine the values of x that satisfy the equation. This method highlights the importance of recognizing the specific structure of the equation and choosing the most appropriate technique for solving it. Understanding the characteristics of the function, such as the absence of the 'bx' term and the value of the constant term, can greatly simplify the solution process and provide valuable insights into the behavior of the function.
Solving for the Roots
To find the roots of f(x) = x² - 48, we set the function equal to zero: x² - 48 = 0. The goal here is to isolate x and find the values that satisfy this equation. This is a fundamental step in solving any equation, as it transforms the problem into a form where the variable can be easily determined. We begin by adding 48 to both sides of the equation: x² = 48. This operation maintains the equality of the equation while moving the constant term to the right side, thus isolating the x² term on the left side. Now, to solve for x, we take the square root of both sides of the equation. It's crucial to remember that taking the square root of a number results in both positive and negative solutions. This is because both the positive and negative square roots, when squared, will yield the original number. Therefore, we have x = ±√48. The ± symbol indicates that there are two possible solutions: one positive and one negative. Next, we simplify the square root of 48. To do this, we look for the largest perfect square that divides 48. The number 16 is the largest perfect square that divides 48 (48 = 16 × 3). We can then rewrite √48 as √(16 × 3). Using the property of square roots that √(ab) = √a × √b, we can separate the square root: √(16 × 3) = √16 × √3. Since √16 = 4, we have √48 = 4√3. Therefore, the roots of the equation are x = ±4√3. This means the function f(x) = x² - 48 intersects the x-axis at two points: x = 4√3 and x = -4√3. These values are the solutions to the equation and represent the x-values for which the function equals zero. The process of solving for the roots involves a series of algebraic manipulations that preserve the equality of the equation while isolating the variable. This step-by-step approach is essential for understanding the underlying principles of equation solving and for applying these techniques to more complex problems.
The Roots: -4√3 and 4√3
Therefore, the roots of the function f(x) = x² - 48 are -4√3 and 4√3. These are the values of x for which the function equals zero, and they represent the x-intercepts of the parabola. The negative root, -4√3, indicates the point where the parabola intersects the x-axis to the left of the y-axis, while the positive root, 4√3, indicates the point where the parabola intersects the x-axis to the right of the y-axis. These roots are symmetrical about the y-axis, which is a direct consequence of the absence of the 'bx' term in the quadratic function. The symmetry of the roots reflects the symmetry of the parabola itself, which is centered around the y-axis. To further understand the significance of these roots, we can approximate their values as decimals. The square root of 3 is approximately 1.732. Therefore, -4√3 is approximately -4 * 1.732 = -6.928, and 4√3 is approximately 4 * 1.732 = 6.928. These decimal approximations provide a clearer sense of the numerical values of the roots and their positions on the number line. The roots are crucial for sketching the graph of the quadratic function. Knowing the roots, the y-intercept (0, -48), and the symmetry about the y-axis, we can accurately sketch the parabola. The parabola opens upwards, passes through the points (-4√3, 0) and (4√3, 0) on the x-axis, and intersects the y-axis at (0, -48). The vertex of the parabola, which is the lowest point on the graph, is located at (0, -48). The roots also play a critical role in various applications of quadratic functions. For example, in physics, they can represent the points where a projectile lands on the ground. In engineering, they can represent the equilibrium points of a system. In economics, they can represent the break-even points of a business. Understanding the roots of a quadratic function is therefore essential for solving a wide range of real-world problems. The process of finding and interpreting the roots highlights the interconnectedness of algebraic and graphical representations of functions, providing a comprehensive understanding of the function's behavior.
The Correct Answer
Based on our calculations, the correct answer is B. -4√3 and 4√3. This option matches the roots we found by setting f(x) = 0 and solving for x. The other options are incorrect because they do not satisfy the equation x² - 48 = 0. For instance, option A, -24 and 24, would imply that (-24)² - 48 = 0 and (24)² - 48 = 0, which are both false. Similarly, options C and D do not yield zero when substituted into the equation. The process of verifying the roots involves substituting the values back into the original equation and checking if the equation holds true. This step is crucial for ensuring the accuracy of the solution and for identifying any potential errors in the calculation. By substituting -4√3 and 4√3 into the equation f(x) = x² - 48, we get: For x = -4√3: f(-4√3) = (-4√3)² - 48 = 16 * 3 - 48 = 48 - 48 = 0 For x = 4√3: f(4√3) = (4√3)² - 48 = 16 * 3 - 48 = 48 - 48 = 0 Since both values satisfy the equation, they are indeed the roots of the function. This verification process reinforces the understanding of what it means for a value to be a root of a function: it is a value that, when substituted into the function, makes the function equal to zero. The correct identification of the roots not only provides the solution to the problem but also deepens the understanding of the behavior and characteristics of the quadratic function. The roots, along with other key features such as the vertex and y-intercept, provide a comprehensive picture of the function's graph and its relationship to the x-axis.
Conclusion
In conclusion, finding the roots of a quadratic function like f(x) = x² - 48 involves setting the function equal to zero and solving for the variable. In this case, the roots are -4√3 and 4√3. Understanding how to find roots is a fundamental skill in mathematics and is essential for solving a variety of problems. The process of finding the roots not only provides the solutions to the equation but also enhances the understanding of the function's behavior and its graphical representation. By setting the function equal to zero, we are essentially finding the x-intercepts of the parabola, which are the points where the graph crosses the x-axis. These points are crucial for sketching the graph of the function and for understanding its overall shape and position. The roots also play a significant role in various applications of quadratic functions in different fields, such as physics, engineering, and economics. In physics, for example, the roots can represent the points where a projectile lands on the ground, or the equilibrium points of a system. In engineering, they can represent the critical values in a design problem. In economics, they can represent the break-even points of a business. The ability to find and interpret the roots of a quadratic function is therefore a valuable skill in many contexts. The methods used to find the roots, such as factoring, completing the square, or using the quadratic formula, provide a toolkit for solving a wide range of quadratic equations. Each method has its own advantages and disadvantages, and the choice of method often depends on the specific characteristics of the equation. In the case of f(x) = x² - 48, the absence of the 'bx' term made the square root property the most efficient method for finding the roots. By mastering these techniques and understanding the underlying principles, one can confidently tackle a variety of mathematical problems and applications involving quadratic functions. The journey from setting up the equation to finding the roots and interpreting their significance is a testament to the power and elegance of mathematics in solving real-world problems.
