Solving 2x^2 = 50 Algebraically A Step-by-Step Guide

Introduction

In this article, we will delve into the algebraic methods for solving the quadratic equation 2x² = 50. Mastering the techniques to solve such equations is crucial in mathematics, as quadratic equations appear in various contexts, from physics and engineering to economics and computer science. Our primary focus will be on isolating the variable x and finding all possible values that satisfy the given equation. We will explore the step-by-step process, providing a clear and comprehensive understanding of the solution. This includes simplifying the equation, applying the square root property, and verifying the solutions. Whether you're a student learning algebra, a teacher looking for a detailed explanation, or simply someone interested in mathematical problem-solving, this guide aims to provide a thorough and accessible approach to solving this type of quadratic equation.

Understanding Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally represented in the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The equation we are addressing, 2x² = 50, is a special case where b is zero, simplifying the solving process. These equations are fundamental in various fields due to their ability to model parabolic relationships. In physics, they describe projectile motion; in engineering, they appear in circuit analysis; and in computer science, they are used in algorithms and optimization problems. Understanding the nature and solutions of quadratic equations is essential for anyone pursuing studies or careers in these areas. The solutions, also known as roots or zeros, can be real or complex, and finding them often involves algebraic manipulation, factoring, completing the square, or using the quadratic formula. Our focus here will be on algebraic manipulation, which is particularly effective for equations of this simplified form.

Step-by-Step Solution

To solve the equation 2x² = 50 algebraically, we will follow a systematic approach that involves simplifying the equation, isolating the variable term, and then finding the square root to determine the values of x. Each step is crucial and builds upon the previous one, ensuring a clear and accurate solution. This method is not only effective for this particular equation but also provides a general strategy for solving similar quadratic equations. Let's break down the solution step-by-step:

Step 1: Simplify the Equation

The initial step in solving the equation 2x² = 50 is to simplify it by isolating the term containing the variable. This involves dividing both sides of the equation by the coefficient of x², which in this case is 2. This step is crucial because it reduces the complexity of the equation and makes it easier to work with. Performing this operation, we get:

2x² / 2 = 50 / 2
x² = 25

This simplification transforms the equation into a more manageable form, where we now have x² isolated on one side. This is a standard technique in algebraic manipulation and is applicable to a wide range of equations. By performing this step, we are setting the stage for the next phase of the solution, which involves finding the values of x that satisfy the simplified equation. The importance of this step cannot be overstated, as it lays the groundwork for a clear and accurate solution.

Step 2: Apply the Square Root Property

Once we have simplified the equation to x² = 25, the next step is to apply the square root property. This property states that if x² = a, then x = ±√a. It's essential to consider both the positive and negative square roots because squaring either value will result in a. This is a fundamental concept in solving quadratic equations and ensures that we find all possible solutions. Applying this property to our equation x² = 25, we get:

x = ±√25

The square root of 25 is 5, so we have:

x = ±5

This means that x can be either +5 or -5. The inclusion of both positive and negative roots is critical in providing a complete solution to the quadratic equation. By understanding and applying the square root property correctly, we ensure that no possible solutions are overlooked. This step is a cornerstone in solving quadratic equations of this form and demonstrates the importance of considering all mathematical possibilities.

Step 3: Verify the Solutions

The final step in solving any equation is to verify the solutions. This involves substituting each value of x back into the original equation to ensure it holds true. Verification is a crucial step because it confirms the accuracy of the solution and helps identify any potential errors made during the solving process. In our case, we found two possible solutions: x = 5 and x = -5. Let's substitute each of these values back into the original equation, 2x² = 50:

For x = 5:

2(5)² = 2(25) = 50

For x = -5:

2(-5)² = 2(25) = 50

Both values satisfy the original equation, confirming that our solutions are correct. Verification is not just a formality; it's an integral part of the problem-solving process. By taking the time to check our work, we can be confident in the accuracy of our solutions. This step reinforces the importance of precision in mathematical problem-solving and ensures a complete and correct answer.

Conclusion

In summary, we have successfully solved the equation 2x² = 50 using algebraic methods. The process involved simplifying the equation by dividing both sides by 2, applying the square root property to find the possible values of x, and verifying these solutions by substituting them back into the original equation. This systematic approach not only provides the correct answers but also enhances understanding of the underlying mathematical principles. The solutions to the equation 2x² = 50 are x = 5 and x = -5. These are the only two values that satisfy the equation, and our verification step confirmed their accuracy. The techniques used in this solution, such as simplifying equations and applying the square root property, are fundamental in algebra and can be applied to a variety of mathematical problems. By mastering these methods, one can confidently tackle similar quadratic equations and build a strong foundation in mathematics.

Understanding and solving quadratic equations like this is crucial for further studies in mathematics and related fields. The ability to manipulate equations algebraically is a valuable skill that extends beyond the classroom and into real-world applications. Whether you are solving physics problems, designing engineering systems, or analyzing economic models, a solid understanding of algebra is essential. This guide has provided a comprehensive walkthrough of the solution, emphasizing the importance of each step and the underlying mathematical principles. By following this approach, anyone can develop the skills necessary to solve quadratic equations and excel in their mathematical endeavors.

Further Practice

To further enhance your understanding and proficiency in solving quadratic equations, it's essential to engage in additional practice. Solving a variety of problems will help you become more comfortable with the different techniques and approaches. Consider working through similar equations with different coefficients and constants. For example, you might try solving 3x² = 75 or x² - 16 = 0. These exercises will reinforce the steps we've discussed and help you identify patterns and shortcuts. Additionally, explore problems that require a combination of algebraic techniques, such as simplifying, factoring, and applying the quadratic formula. This will provide a more comprehensive understanding of quadratic equations and their solutions.

Another valuable practice is to apply these skills to real-world scenarios. Many practical problems in physics, engineering, and economics can be modeled using quadratic equations. For example, you might calculate the trajectory of a projectile, determine the dimensions of a rectangle with a given area, or analyze the growth of a population. By connecting the mathematical concepts to real-world applications, you can gain a deeper appreciation for the power and versatility of quadratic equations. Furthermore, consider seeking out online resources, textbooks, and practice problems to supplement your learning. There are numerous tools and materials available to help you master this important topic in algebra. The key is consistent practice and a willingness to explore different types of problems. With dedication and effort, you can develop the skills and confidence needed to solve quadratic equations effectively and efficiently.