Solving The Quadratic Equation 50 - X^2 = 0

In this article, we will explore how to find the solution(s) to the quadratic equation 50 - x² = 0. Quadratic equations are a fundamental part of algebra, and understanding how to solve them is crucial for various mathematical and real-world applications. This equation is a simple yet illustrative example of a quadratic equation that can be solved using several methods. We will walk through the steps to find the solution(s), discuss the underlying concepts, and clarify why a particular answer is correct. Our focus will be on ensuring clarity and providing a comprehensive explanation so that anyone, regardless of their mathematical background, can follow along and understand the process. The correct solution will be identified from the given options, and a detailed explanation will support our choice. Mastering quadratic equations opens doors to solving more complex problems in physics, engineering, economics, and computer science. So, let's dive in and demystify this mathematical concept.

To effectively solve the equation 50 - x² = 0, it's important to first understand what a quadratic equation is and its general form. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in this case, x) is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. In our equation, 50 - x² = 0, we can rewrite it as -x² + 50 = 0 to better see its quadratic nature. Here, a = -1, b = 0 (since there is no x term), and c = 50.

Solving a quadratic equation means finding the values of x that make the equation true. These values are also known as the roots or solutions of the equation. There are several methods to solve quadratic equations, including factoring, using the quadratic formula, completing the square, and in simpler cases, isolating the variable. The method we choose often depends on the specific form of the equation. For instance, when there is no x term (i.e., b = 0), as in our equation, isolating the variable is usually the most straightforward approach. This involves rearranging the equation to get x² on one side and a constant on the other, and then taking the square root of both sides. This understanding sets the stage for solving our specific equation, which we will tackle step by step in the next section.

Now, let's solve the quadratic equation 50 - x² = 0 step by step. Our goal is to isolate x and find the values that satisfy the equation. First, we can rearrange the equation to get the x² term on one side:

50 - x² = 0

Add x² to both sides:

50 = x²

This rearrangement simplifies the equation significantly. Now, we have x² isolated on one side. The next step is to take the square root of both sides of the equation to solve for x. Remember, when taking the square root, we must consider both the positive and negative roots because both positive and negative numbers, when squared, can yield a positive result. So, we have:

√x² = ±√50

This gives us:

x = ±√50

Now, we need to simplify √50. We can do this by factoring 50 into its prime factors or recognizing perfect square factors. 50 can be factored as 2 × 25, where 25 is a perfect square (5²). Thus, we can rewrite √50 as √(25 × 2). Using the property of square roots that √(a × b) = √a × √b, we get:

√50 = √25 × √2 = 5√2

Substituting this back into our equation, we have:

x = ±5√2

Therefore, the solutions to the quadratic equation 50 - x² = 0 are x = 5√2 and x = -5√2. This detailed step-by-step solution clarifies how we arrive at the final answer, making it easy to follow along and understand the process.

Having solved the quadratic equation 50 - x² = 0, we found the solutions to be x = ±5√2. Now, we need to identify which of the given options matches our solution. Let's revisit the options:

A. x = ±2√5 B. x = ±6√3 C. x = ±5√2 D. no real solution

By comparing our solution x = ±5√2 with the given options, it's clear that option C, x = ±5√2, is the correct answer. The other options do not match our calculated solutions. Option A, x = ±2√5, has different coefficients and radicands. Option B, x = ±6√3, similarly has different values. Option D, stating "no real solution," is incorrect because we have clearly found two real solutions.

This process of comparing the derived solutions with the given options is a crucial step in problem-solving. It ensures that the final answer aligns with the question's requirements and the provided choices. In this case, the correct identification of option C as the solution confirms our step-by-step calculation and demonstrates the accuracy of our method. It reinforces the importance of careful calculation and comparison to arrive at the correct answer in mathematical problems.

While we solved the equation 50 - x² = 0 by isolating the variable and taking the square root, it's worth noting that quadratic equations can be solved using other methods as well. Understanding these alternative methods can provide a broader perspective and can be useful for solving different types of quadratic equations. One common method is the quadratic formula. The quadratic formula is a general solution that can be used for any quadratic equation in the form ax² + bx + c = 0. The formula is:

x = [-b ± √(b² - 4ac)] / (2a)

In our equation, 50 - x² = 0, we have a = -1, b = 0, and c = 50. Plugging these values into the quadratic formula, we get:

x = [-0 ± √(0² - 4(-1)(50))] / (2(-1)) x = [± √(200)] / (-2) x = [± 10√2] / (-2) x = ± 5√2

As we can see, the quadratic formula also yields the same solutions, x = ±5√2. This demonstrates the versatility of the quadratic formula as a method for solving quadratic equations. Another method, completing the square, is also a powerful technique, especially for equations that are not easily factored. However, for this specific equation, isolating the variable and the quadratic formula are more straightforward. Understanding these various methods allows for a flexible approach to solving quadratic equations, making it easier to tackle different problems.

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications in various fields. Understanding how to solve them can provide insights into many practical problems. One common application is in physics, particularly in the study of projectile motion. For example, the trajectory of a ball thrown into the air can be modeled using a quadratic equation. The equation can help determine the height of the ball at any given time or the distance it will travel before hitting the ground. In engineering, quadratic equations are used in the design of bridges, buildings, and other structures. Engineers use these equations to calculate stresses, strains, and deflections, ensuring the stability and safety of their designs.

In economics and finance, quadratic equations can be used to model cost, revenue, and profit functions. Businesses can use these models to determine the optimal pricing strategy or production levels to maximize profits. Computer graphics also utilizes quadratic equations to create curves and surfaces. The smooth curves seen in computer-aided design (CAD) software and video games often rely on quadratic functions. Furthermore, quadratic equations are used in optimization problems, where the goal is to find the maximum or minimum value of a function. This is crucial in various fields, including logistics, resource allocation, and machine learning. The ability to solve quadratic equations, therefore, extends beyond the classroom and becomes a valuable tool in many professional and practical contexts.

When solving quadratic equations, several common mistakes can lead to incorrect answers. Recognizing these pitfalls and understanding how to avoid them is crucial for accuracy. One frequent mistake is forgetting to consider both positive and negative roots when taking the square root of a number. For example, when solving x² = 25, students might only think of the solution x = 5, neglecting x = -5. Remember that both 5² and (-5)² equal 25, so both are valid solutions. Another common error occurs when applying the quadratic formula. Incorrectly substituting values for a, b, or c, or making mistakes in the arithmetic, can lead to wrong solutions. It's essential to carefully write out each step and double-check the substitutions and calculations.

Another pitfall is misinterpreting the structure of the equation. For instance, in the equation 50 - x² = 0, some might mistakenly try to factor it as a difference of squares without proper rearrangement. It's crucial to rearrange the equation to the standard form or a more manageable format before applying a solution method. Additionally, sign errors are common, especially when dealing with negative coefficients or constants. Paying close attention to the signs in each step of the process is vital. To avoid these mistakes, practice is key. Working through a variety of quadratic equations and checking the solutions can help reinforce the correct methods and build confidence. Additionally, using alternative methods to solve the same equation can serve as a verification step, ensuring the accuracy of the final answer.

In conclusion, solving the quadratic equation 50 - x² = 0 involves understanding the fundamental properties of quadratic equations and applying appropriate solution methods. We systematically isolated the variable x, took the square root of both sides, and simplified the result to find the solutions x = ±5√2. This process demonstrates a straightforward approach to solving quadratic equations when the b term is zero. We also explored alternative methods such as the quadratic formula, which confirmed our solutions and highlighted the versatility of different techniques. Identifying the correct answer from the given options reinforced the importance of careful comparison and verification.

Furthermore, we discussed the real-world applications of quadratic equations across various fields, illustrating their practical relevance beyond academic exercises. Understanding these applications helps to appreciate the significance of mastering quadratic equations. Common mistakes in solving these equations were also addressed, along with strategies to avoid them, emphasizing the need for meticulous calculation and a thorough understanding of the underlying concepts. By mastering quadratic equations, one gains a valuable tool for problem-solving in mathematics and various real-world contexts. The ability to confidently solve these equations opens doors to more advanced mathematical concepts and applications, making it a crucial skill in STEM fields and beyond. Thus, a solid grasp of quadratic equations is an investment in one's mathematical proficiency and problem-solving capabilities.