Solving Equations Involving x and 1/x: A Comprehensive Guide

When dealing with equations that involve the variable x and 1/x, we often stumble upon interesting algebraic manipulations. These equations frequently arise in various mathematical problems, particularly in advanced algebra and calculus. This guide will explore how to solve a specific type of equation, namely those involving expressions like x(1/x). We will delve into the algebraic manipulations and derive solutions, illustrating intricate problem-solving techniques with detailed steps.

Introduction to x and 1/x Equations

The equation x(1/x) k, where k is a constant, forms a basis for many algebraic challenges. This simple expression can lead to a variety of solutions, especially when higher powers and more complex operations are involved. For instance, consider the given equation:

left( frac{1}{x} right)^2 3 implies x cdot frac{1}{x} sqrt{3}

Step-by-Step Solution

Given the equation:

Start with the equation left( frac{1}{x} right)^2 3 Take the square root of both sides: frac{1}{x} sqrt{3} Multiply both sides by x to get: x cdot frac{1}{x} sqrt{3} Note that x cdot frac{1}{x} 1 (for non-zero x), so the equation simplifies to: 1 sqrt{3}, which is not true. This indicates a need for a different approach.

Instead, let's proceed with:

Revisit the original equation: left( frac{1}{x} right)^2 3 Multiply both sides by x^4, yielding: 1 3x^2 Solving for x^2, we get: x^2 frac{1}{3} Therefore, x pm frac{1}{sqrt{3}}

For the next equation:

left( frac{1}{x} right)^3 3sqrt{3} Multiply both sides by x^6, yielding: 1 3sqrt{3}x^3 Solving for x^3, we obtain: x^3 frac{1}{3sqrt{3}} Therefore, x left( frac{1}{3sqrt{3}} right)^{1/3}

Alternative Approaches

Let's explore another method:

Given the equation x cdot frac{1}{x} sqrt{2} Square both sides to get: left( x cdot frac{1}{x} right)^2 2 This simplifies to: x^2 - 1 2 Rearrange to find: x^2 3 Therefore, x pm sqrt{3}

For a more intricate approach:

Given the equation x cdot frac{1}{x} sqrt{2} Cube both sides: left( x cdot frac{1}{x} right)^3 2^{3/2} This results in: x^3 - x 3sqrt{2} Rearrange to: x^3 - 3sqrt{2}x - 1 0

Derivation of 1/x^3

Given the equation x cdot frac{1}{x} sqrt{2}:

Define u x^3 The equation transforms to: u frac{1}{u} -sqrt{2} Multiply by u to get: u^2 1 -sqrt{2}u Apply the quadratic formula: u frac{-sqrt{2}pm sqrt{2 4}}{2} This simplifies to: u frac{-sqrt{2}}{2}(1pm i) Therefore, x^3 u^{1/3} left( frac{-sqrt{2}}{2}(1pm i) right)^{1/3}

Conclusions

The solutions for 1/x^3 can be derived as:

1/x^3 -i 1/x^3 1/i^3 -i

This concludes our exploration into solving equations involving x and 1/x. Understanding these techniques is essential for tackling more complex algebraic problems and enhances problem-solving skills in general.