Solving Algebraic Equations: x - 1/x 4 and Evaluating x2 - 1/x2 and x? - 1/x?

Introduction:

Algebraic equations form the backbone of many advanced mathematical concepts. This article explores the process of solving a specific algebraic equation involving square terms. We begin with the equation x - 1/x 4 and proceed to find the values of x2 - 1/x2 and x? - 1/x?. The steps and methods involved in solving these equations are explained in detail, providing insights into how to manipulate and simplify expressions.

Solving for x - 1/x 4

Given the equation x - 1/x 4, we start by squaring both sides to eliminate the fraction.

Step 1: Squaring Both Sides

[x - frac{1}{x} 4] [left(x - frac{1}{x}right)^2 4^2] [x^2 - 2 cdot x cdot frac{1}{x} frac{1}{x^2} 16] [x^2 - 2 frac{1}{x^2} 16] [x^2 frac{1}{x^2} 16 2] [x^2 frac{1}{x^2} 18]

Step 2: Squaring the New Expression

Next, we square the equation x2 1/x2 18 to find x? - 1/x?.

[left(x^2 frac{1}{x^2}right)^2 18^2] [x^4 2 cdot x^2 cdot frac{1}{x^2} frac{1}{x^4} 324] [x^4 2 frac{1}{x^4} 324] [x^4 frac{1}{x^4} 324 - 2] [x^4 frac{1}{x^4} 322]

Complete Solution

Given the equation x - 1/x 4, we proceed to solve it step by step.

Step 1: Squaring the Equation

[x - frac{1}{x} 4]

First, square both sides:

[left(x - frac{1}{x}right)^2 4^2]

Expanding the left side:

[x^2 - 2 cdot x cdot frac{1}{x} frac{1}{x^2} 16]

Simplifying:

[x^2 - 2 frac{1}{x^2} 16]

Combining like terms:

[x^2 frac{1}{x^2} 18]

Step 2: Squaring the New Expression

Now, we square the expression x2 1/x2 to find x? - 1/x?.

[left(x^2 frac{1}{x^2}right)^2 18^2]

Expanding the left side:

[x^4 2 cdot x^2 cdot frac{1}{x^2} frac{1}{x^4} 324]

Simplifying:

[x^4 2 frac{1}{x^4} 324]

Combining like terms:

[x^4 frac{1}{x^4} 322]

Thus, the final values are:

x2 - 1/x2 18 x? - 1/x? 322

This solution method can be extended to other similar equations, providing a robust framework for solving complex algebraic problems.