Solving For X - 1/x Given 3(x^2 + 1/x^2) + 10(x - 1/x) + 1 = 0
Solve for x - 1/x if 3(x^2 + 1/x^2) + 10(x - 1/x) + 1 = 0. Options: -1 or 1, -7/3 or 7/3, -1 or -7/3, -1 or 2/3.
Introduction
In this article, we will delve into the process of solving a specific algebraic equation to determine the possible values of the expression x - 1/x. The given equation is 3(x^2 + 1/x^2) + 10(x - 1/x) + 1 = 0. This type of equation often appears in mathematical problem-solving scenarios, especially in algebra and calculus. By employing appropriate algebraic techniques and substitutions, we can transform the equation into a more manageable form, ultimately leading us to the desired solutions. Our primary goal is to methodically explain the steps involved, ensuring a clear and comprehensive understanding of the solution process. The problem is a classic example of how algebraic manipulation and substitution can simplify complex expressions and lead to elegant solutions.
Understanding the Problem
Before we start solving, let’s take a closer look at the equation 3(x^2 + 1/x^2) + 10(x - 1/x) + 1 = 0. It's an equation involving both quadratic terms (x^2 + 1/x^2) and linear terms (x - 1/x). The presence of both terms suggests a possible substitution method to simplify the equation. By recognizing the relationship between these terms, we can introduce a new variable that will transform the equation into a more familiar quadratic form. This approach not only simplifies the algebraic manipulations but also provides a structured way to find the solutions. The key here is to identify the common structures within the equation that allow for effective substitutions. Moreover, this problem highlights the importance of pattern recognition in mathematical problem-solving. By noticing the interplay between the terms, we can devise an efficient strategy to find the solutions.
Substitution Method
The core of our solution strategy lies in the method of substitution. Let's set y = x - 1/x. Our goal is to express the given equation in terms of y. To do this, we need to find an expression for x^2 + 1/x^2 in terms of y. By squaring both sides of the equation y = x - 1/x, we get y^2 = (x - 1/x)^2. Expanding the right side, we have y^2 = x^2 - 2 + 1/x^2. Rearranging this, we find x^2 + 1/x^2 = y^2 + 2. This substitution is crucial because it allows us to rewrite the original equation in terms of a single variable, thereby simplifying the problem significantly. The ability to recognize and apply appropriate substitutions is a fundamental skill in solving algebraic equations. This technique transforms a seemingly complex equation into a more standard form, making it easier to solve. The elegance of this method lies in its ability to reduce the complexity of the problem by introducing a new variable that captures the underlying structure of the equation.
Transforming the Equation
Now, we substitute y and y^2 + 2 into the original equation 3(x^2 + 1/x^2) + 10(x - 1/x) + 1 = 0. Replacing x^2 + 1/x^2 with y^2 + 2 and x - 1/x with y, we get: 3(y^2 + 2) + 10y + 1 = 0. Expanding and simplifying this equation, we obtain a quadratic equation in terms of y. This step is critical as it transforms the original equation into a familiar quadratic form, which we can easily solve using standard methods. The transformation highlights the power of algebraic manipulation in simplifying complex equations. By expressing the original equation in terms of y, we have effectively reduced the problem to solving a simple quadratic equation. This not only makes the solution process more straightforward but also provides a clear path to finding the values of x - 1/x. The quadratic equation we obtain is a stepping stone to the final solution, and it demonstrates the effectiveness of the substitution method in action.
Solving the Quadratic Equation
Let’s simplify the equation we obtained in the previous step: 3(y^2 + 2) + 10y + 1 = 0. Expanding and combining like terms, we get: 3y^2 + 6 + 10y + 1 = 0, which simplifies to 3y^2 + 10y + 7 = 0. Now, we have a standard quadratic equation in the form ay^2 + by + c = 0. To solve this quadratic equation, we can use factoring, the quadratic formula, or completing the square. In this case, factoring seems to be the most straightforward approach. We look for two numbers that multiply to 3 * 7 = 21 and add up to 10. These numbers are 3 and 7. So, we can rewrite the middle term as 10y = 3y + 7y. This gives us: 3y^2 + 3y + 7y + 7 = 0. Factoring by grouping, we get: 3y(y + 1) + 7(y + 1) = 0. Factoring out the common term (y + 1), we have: (3y + 7)(y + 1) = 0. This gives us two possible solutions for y: 3y + 7 = 0 or y + 1 = 0. Solving these, we find y = -7/3 or y = -1. These values of y are the solutions to the transformed quadratic equation and are critical in finding the final answer. The ability to solve quadratic equations is a fundamental skill in algebra, and this step demonstrates its importance in solving more complex problems. The factoring technique used here is a classic method for solving quadratic equations and provides a clear and efficient way to find the roots.
Finding x - 1/x
Recall that we made the substitution y = x - 1/x. Now that we have the values of y, we can find the possible values of x - 1/x. We found that y = -7/3 or y = -1. Therefore, x - 1/x = -7/3 or x - 1/x = -1. These are the solutions to the original problem. We have successfully found the possible values of the expression x - 1/x by using a combination of algebraic manipulation and substitution. This step concludes the solution process and provides the final answer to the problem. The ability to trace back the substitutions and find the values of the original variables is a crucial skill in problem-solving. This step highlights the importance of keeping track of the substitutions made and ensuring that the final answer is in terms of the original variables. The solutions x - 1/x = -7/3 and x - 1/x = -1 are the final results of our methodical approach.
Conclusion
In this article, we have demonstrated a step-by-step solution to find the values of x - 1/x when 3(x^2 + 1/x^2) + 10(x - 1/x) + 1 = 0. We used the substitution method to transform the equation into a quadratic equation, solved the quadratic equation, and then found the values of x - 1/x. The solutions are x - 1/x = -7/3 or x - 1/x = -1. This problem illustrates the power of algebraic techniques and substitution in solving complex equations. By breaking down the problem into smaller, manageable steps, we were able to find the solutions efficiently. The process involved identifying the key relationships between the terms in the equation, making an appropriate substitution, solving the resulting quadratic equation, and finally, finding the values of the desired expression. This methodical approach is a valuable skill in mathematics and can be applied to a wide range of problems. The solutions obtained are not just numerical answers but also represent a deeper understanding of the underlying algebraic structures. The ability to solve such problems is a testament to the power of algebraic manipulation and the beauty of mathematical problem-solving.