Solving X²+3x-(x-2)=2(x-1)+(x²-x) A Step-by-Step Mathematical Discussion
In the realm of mathematics, equations serve as the fundamental building blocks for understanding and describing the relationships between various quantities. Solving equations allows us to determine the values of unknown variables, providing insights into the underlying mathematical structures. Today, we delve into the equation x²+3x-(x-2)=2(x-1)+(x²-x), a seemingly complex expression that, upon closer inspection, reveals a straightforward solution. This article aims to provide a comprehensive discussion of the equation, guiding you through the steps involved in simplifying and solving it, while also emphasizing the underlying mathematical principles.
1. The Initial Equation: A Complex Expression
The equation we aim to solve is: x²+3x-(x-2)=2(x-1)+(x²-x). At first glance, the equation might appear daunting, with its mix of quadratic terms, linear terms, and parentheses. However, with a systematic approach, we can unravel its complexity and arrive at a solution. The initial step involves simplifying both sides of the equation by expanding the parentheses and combining like terms. This process will help us to eliminate the visual clutter and reveal the underlying structure of the equation. When simplifying, it is crucial to pay close attention to the signs and coefficients of each term to ensure accuracy. A single mistake in the simplification process can lead to an incorrect solution. Therefore, taking our time and double-checking each step is essential for success. Before diving into the simplification, let's take a moment to appreciate the elegance of mathematical equations. They provide a concise and precise way to express complex relationships, allowing us to model and understand the world around us. Solving equations is not just about finding the numerical answer; it's about developing our logical reasoning skills and enhancing our understanding of mathematical concepts. So, with that in mind, let's begin our journey of unraveling this equation.
2. Simplifying the Equation: Unveiling the Structure
Simplifying equations is an essential step in solving mathematical problems. Our initial equation, x²+3x-(x-2)=2(x-1)+(x²-x), requires us to simplify both sides to isolate the variable 'x'. The first step in simplifying the equation is to remove the parentheses. On the left side, we have -(x-2), which means we need to distribute the negative sign to both terms inside the parentheses. This gives us -x + 2. On the right side, we have 2(x-1), which means we need to multiply both terms inside the parentheses by 2. This gives us 2x - 2. So, after removing the parentheses, the equation becomes: x²+3x-x+2=2x-2+x²-x. Now, we need to combine like terms on both sides of the equation. On the left side, we have x² term, two x terms (3x and -x), and a constant term (2). Combining the x terms gives us 3x - x = 2x. So, the left side of the equation becomes: x² + 2x + 2. On the right side, we also have an x² term, two x terms (2x and -x), and a constant term (-2). Combining the x terms gives us 2x - x = x. So, the right side of the equation becomes: x² + x - 2. Now, our simplified equation looks like this: x² + 2x + 2 = x² + x - 2. Notice how much simpler the equation looks now compared to the initial form. By carefully removing the parentheses and combining like terms, we have unveiled the underlying structure of the equation, making it easier to solve. In the next step, we will further simplify the equation by moving all the terms to one side, which will help us to isolate the variable 'x' and find its value. Remember, the goal of simplification is to make the equation more manageable and easier to work with. So, always take your time and double-check each step to avoid errors. By mastering the art of simplification, you will be well-equipped to tackle more complex mathematical problems.
3. Isolating the Variable: Bringing Terms Together
To isolate the variable x and solve the equation x² + 2x + 2 = x² + x - 2, we need to bring all the terms to one side of the equation. This is achieved by performing the same operations on both sides, ensuring the equation remains balanced. Let's start by subtracting x² from both sides of the equation. This eliminates the x² term from both sides, simplifying the equation further. Subtracting x² from both sides, we get: x² + 2x + 2 - x² = x² + x - 2 - x². This simplifies to: 2x + 2 = x - 2. Now, we need to isolate the x term. To do this, let's subtract x from both sides of the equation. This will move the x term from the right side to the left side. Subtracting x from both sides, we get: 2x + 2 - x = x - 2 - x. This simplifies to: x + 2 = -2. We are now very close to isolating x. The final step is to subtract 2 from both sides of the equation. This will eliminate the constant term on the left side and leave us with x isolated on one side. Subtracting 2 from both sides, we get: x + 2 - 2 = -2 - 2. This simplifies to: x = -4. Therefore, the solution to the equation is x = -4. We have successfully isolated the variable x by systematically performing operations on both sides of the equation. This process of isolating the variable is a fundamental technique in algebra and is used to solve a wide variety of equations. By understanding how to isolate the variable, you can confidently tackle more complex equations and mathematical problems. In the next section, we will verify our solution by substituting it back into the original equation to ensure its correctness.
4. Verifying the Solution: Ensuring Accuracy
Verifying the solution is a crucial step in the problem-solving process. To ensure that our solution, x = -4, is correct, we must substitute this value back into the original equation: x²+3x-(x-2)=2(x-1)+(x²-x). Substituting x = -4 into the equation, we get: (-4)² + 3(-4) - ((-4) - 2) = 2((-4) - 1) + ((-4)² - (-4)). Now, we need to simplify both sides of the equation to see if they are equal. Let's start with the left side. (-4)² is 16. 3(-4) is -12. (-4) - 2 is -6, and subtracting -6 is the same as adding 6. So, the left side becomes: 16 - 12 + 6. Simplifying this, we get: 16 - 12 = 4, and 4 + 6 = 10. Therefore, the left side of the equation simplifies to 10. Now, let's simplify the right side of the equation. (-4) - 1 is -5. 2(-5) is -10. (-4)² is 16, and subtracting -4 is the same as adding 4. So, the right side becomes: -10 + (16 + 4). Simplifying this, we get: 16 + 4 = 20, and -10 + 20 = 10. Therefore, the right side of the equation also simplifies to 10. Since both sides of the equation simplify to 10, we have verified that our solution, x = -4, is correct. Verification is a valuable tool for catching errors and ensuring the accuracy of our solutions. By substituting the solution back into the original equation, we can confirm that it satisfies the equation and that we have not made any mistakes in our calculations. Always remember to verify your solutions, especially in complex problems, to avoid making careless errors. In this case, we have successfully verified our solution and can confidently conclude that x = -4 is the correct answer to the equation x²+3x-(x-2)=2(x-1)+(x²-x).
5. Conclusion: The Solution and Its Significance
In conclusion, we have successfully unraveled the equation x²+3x-(x-2)=2(x-1)+(x²-x). Through a systematic process of simplification, isolating the variable, and verification, we have determined that the solution to the equation is x = -4. This journey through the equation has not only provided us with a numerical answer but has also reinforced our understanding of fundamental algebraic principles. The importance of simplification cannot be overstated. By carefully expanding parentheses, combining like terms, and rearranging the equation, we transformed a seemingly complex expression into a manageable form. This highlights the power of breaking down complex problems into smaller, more digestible steps. Isolating the variable is a core technique in algebra, allowing us to determine the value of the unknown. By performing the same operations on both sides of the equation, we maintained balance while gradually isolating x. This process demonstrates the importance of maintaining equality in mathematical manipulations. Verification, the final step, is often overlooked but is crucial for ensuring the accuracy of our solution. By substituting x = -4 back into the original equation, we confirmed that it satisfied the equation, giving us confidence in our answer. The solution x = -4 represents a specific point on the number line that, when substituted into the equation, makes both sides equal. This concept is fundamental to understanding the nature of equations and their solutions. In a broader context, solving equations is a fundamental skill in mathematics and has applications in various fields, including science, engineering, economics, and computer science. The ability to manipulate equations and solve for unknowns is essential for modeling real-world phenomena, making predictions, and solving problems. Therefore, mastering the techniques discussed in this article will not only help you solve this particular equation but will also equip you with valuable skills for tackling a wide range of mathematical challenges. As you continue your mathematical journey, remember the importance of simplification, isolation, and verification. These principles will serve as your guide in navigating the world of equations and beyond.