Solving The Equation 1 = 1/(x^2 + 2x) + (x - 1)/x A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. Equations are mathematical statements that assert the equality of two expressions. Solving an equation means finding the value(s) of the variable(s) that make the equation true. This article delves into the intricacies of solving the equation 1 = 1/(x^2 + 2x) + (x - 1)/x, offering a step-by-step guide and exploring the underlying mathematical concepts. Let's embark on this mathematical journey to master the art of equation solving and appreciate the elegance of algebraic manipulations. The process of solving equations is not merely about finding answers; it's about understanding the relationships between variables and constants, and the logical steps required to isolate the unknown. This equation, while seemingly simple, involves fractions and quadratic expressions, providing a rich opportunity to enhance our problem-solving skills. By the end of this comprehensive guide, you'll have a firm grasp on the techniques needed to tackle similar algebraic challenges with confidence. Equations form the backbone of numerous scientific and engineering disciplines, making their mastery essential for anyone pursuing these fields. Furthermore, the logical thinking and problem-solving skills honed through equation solving are invaluable in everyday life, fostering a structured approach to challenges and decision-making. So, let's dive in and unlock the secrets hidden within this equation.
Step 1: Identifying the Domain and Restrictions
Before we begin manipulating the equation, it's crucial to identify the domain of the variable x. The domain is the set of all possible values that x can take without making the equation undefined. In this case, we have fractions in the equation, which means we need to avoid values of x that would make the denominators equal to zero. Specifically, we need to ensure that x^2 + 2x ≠0 and x ≠0. Let's analyze these conditions:
- x^2 + 2x ≠0 can be factored as x(x + 2) ≠0. This implies that x ≠0 and x ≠-2.
- x ≠0 is already a condition we've identified.
Therefore, the domain of x is all real numbers except 0 and -2. These values are critical restrictions because they would lead to division by zero, which is undefined in mathematics. Recognizing these restrictions early on is essential to ensure that our solutions are valid and don't fall outside the permissible range. It's a common mistake to solve an equation without considering the domain, which can lead to extraneous solutions that don't satisfy the original equation. By meticulously identifying the domain at the outset, we lay a solid foundation for accurate and meaningful problem-solving. This step is not just a formality; it's a vital safeguard against errors and a testament to the rigor of mathematical thinking. In more complex equations involving radicals, logarithms, or trigonometric functions, the process of determining the domain becomes even more critical, as these functions have inherent restrictions on their inputs. Hence, mastering the art of domain identification is a cornerstone of mathematical proficiency.
Step 2: Clearing the Fractions
To simplify the equation and make it easier to solve, our next step is to clear the fractions. We achieve this by multiplying both sides of the equation by the least common denominator (LCD) of the fractions. In this case, the denominators are x^2 + 2x and x. Factoring x^2 + 2x gives us x(x + 2), so the LCD is x(x + 2). Now, let's multiply both sides of the equation by x(x + 2):
1 * x(x + 2) = [1/(x^2 + 2x) + (x - 1)/x] * x(x + 2)
This simplifies to:
x(x + 2) = 1 + (x - 1)(x + 2)
This step is crucial because it transforms the equation from one involving fractions to a simpler polynomial equation. Clearing fractions eliminates the complexity of dealing with denominators and allows us to focus on the algebraic manipulations needed to isolate the variable x. The LCD serves as the common ground that allows us to combine the terms and eliminate the fractional form. It's important to choose the correct LCD to avoid unnecessary complexity in the resulting equation. In more complex equations, finding the LCD might involve factoring polynomials or identifying common factors among multiple denominators. However, the principle remains the same: multiply both sides by the LCD to clear the fractions and pave the way for a more straightforward solution process. This technique is a fundamental tool in algebraic manipulation and is widely applicable in various mathematical contexts.
Step 3: Expanding and Simplifying
Now that we've cleared the fractions, we have a polynomial equation. The next step is to expand the expressions and simplify the equation by combining like terms. Let's expand the right side of the equation: x(x + 2) = 1 + (x - 1)(x + 2).
The left side remains x(x + 2), which expands to x^2 + 2x.
The right side expands as follows:
1 + (x - 1)(x + 2) = 1 + (x^2 + 2x - x - 2) = 1 + x^2 + x - 2 = x^2 + x - 1
Now, our equation looks like this:
x^2 + 2x = x^2 + x - 1
This simplification step is essential for bringing the equation into a manageable form. By expanding the products and combining like terms, we reduce the complexity of the equation and make it easier to identify the next steps in the solution process. It's like decluttering a workspace to create a clearer path forward. In this case, we've transformed a potentially daunting equation into a much simpler one that is readily solvable. The expansion of products involves applying the distributive property and carefully tracking the signs of the terms. Combining like terms requires identifying terms with the same variable and exponent and adding or subtracting their coefficients. These are fundamental algebraic skills that are crucial for success in equation solving and other mathematical endeavors. A meticulous approach to expansion and simplification minimizes the risk of errors and ensures that the resulting equation accurately reflects the original problem.
Step 4: Isolating the Variable
With the equation simplified to x^2 + 2x = x^2 + x - 1, our goal is to isolate the variable x. To do this, we can subtract x^2 from both sides of the equation:
x^2 + 2x - x^2 = x^2 + x - 1 - x^2
This simplifies to:
2x = x - 1
Next, subtract x from both sides:
2x - x = x - 1 - x
This gives us:
x = -1
This isolation step is the heart of the equation-solving process. It involves strategically applying inverse operations to both sides of the equation to gradually peel away the terms surrounding the variable until it stands alone. In this case, we used subtraction to eliminate the x^2 term and then to isolate x on one side of the equation. The key is to maintain the balance of the equation by performing the same operation on both sides. Each step brings us closer to the solution, revealing the value of x that satisfies the original equation. In more complex equations, the isolation process might involve a series of steps, including addition, subtraction, multiplication, division, and even taking roots or logarithms. However, the underlying principle remains the same: use inverse operations to systematically isolate the variable and uncover its value. This skill is fundamental to algebra and is essential for solving a wide range of mathematical problems.
Step 5: Verifying the Solution
We've found a potential solution: x = -1. However, it's crucial to verify that this solution is valid by substituting it back into the original equation. This step ensures that our solution satisfies the equation and that it doesn't violate any restrictions on the domain. Let's substitute x = -1 into the original equation:
1 = 1/((-1)^2 + 2(-1)) + ((-1) - 1)/(-1)
Simplifying the equation:
1 = 1/(1 - 2) + (-2)/(-1)
1 = 1/(-1) + 2
1 = -1 + 2
1 = 1
The equation holds true, so x = -1 is a valid solution.
This verification step is a critical safeguard against errors and extraneous solutions. It's like a final checkmark that confirms the correctness of our work. Extraneous solutions can arise when we perform operations that introduce new solutions that don't satisfy the original equation, such as squaring both sides or multiplying by an expression that could be zero. By substituting our solution back into the original equation, we ensure that it truly satisfies the given conditions. Furthermore, we also need to check if our solution violates any restrictions on the domain. In this case, x = -1 is within the domain we identified earlier (all real numbers except 0 and -2), so it's a valid solution. The verification step reinforces the rigor of mathematical problem-solving and provides confidence in the accuracy of our results. It's a habit that should be cultivated in all mathematical endeavors.
Conclusion: Mastering Equation Solving
In this comprehensive guide, we've meticulously solved the equation 1 = 1/(x^2 + 2x) + (x - 1)/x. We started by identifying the domain and restrictions, cleared the fractions, expanded and simplified the equation, isolated the variable, and finally verified our solution. The solution we found is x = -1. This process highlights the importance of a systematic approach to equation solving, emphasizing the need to consider the domain, manipulate the equation algebraically, and verify the solution. Solving equations is a fundamental skill in mathematics, with applications spanning various fields. By mastering the techniques outlined in this guide, you'll be well-equipped to tackle a wide range of algebraic challenges. Remember, the key to success lies in understanding the underlying concepts, practicing diligently, and approaching problems with a clear and organized mindset. Equation solving is not just about finding answers; it's about developing logical thinking, problem-solving skills, and a deep appreciation for the elegance and power of mathematics. As you continue your mathematical journey, remember to embrace challenges, learn from mistakes, and celebrate the joy of discovery that comes with unraveling the mysteries of equations.
