Understanding the Equation ( x^2 - frac{1}{x^2} left(x - frac{1}{x}right)^2 - 2 ) and Its Solutions

In this article, we delve into the equation ( x^2 - frac{1}{x^2} left(x - frac{1}{x}right)^2 - 2 ). We will explore the algebraic manipulation required to understand the underlying identity and determine the set of solutions.

Algebraic Manipulation and Simplification

First, let's start by examining the right-hand side (RHS) of the equation:

Step 1: Expand the RHS

The RHS of the equation is given by:

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

Expanding the square term, we get:

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

Subtracting 2 from this expression, we have:

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

Simplifying, we get:

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

which is exactly the left-hand side (LHS) of the equation.

Conclusion

Hence, the equation ( x^2 - frac{1}{x^2} left(x - frac{1}{x}right)^2 - 2 ) simplifies to:

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

This identity is true for all real numbers ( x ), except when ( x 0 ), because the original equation involves division by ( x ).

Solution Set

The solution set for the equation is:

[boxed{x eq 0}]

or equivalently:

[boxed{left(-infty, 0 cup (0, infty)right)}]

This means that the equation holds for all real numbers excluding 0.

Explanation of the Solution Set

Let's break down why ( x eq 0 ) is the solution set:

1. **Left Side**: The term ( x^2 - frac{1}{x^2} ) is defined for all real numbers except 0. When ( x 0 ), the term ( frac{1}{x^2} ) is undefined.

2. **Right Side**: The term ( left(x - frac{1}{x}right)^2 - 2 ) is also undefined when ( x 0 ), because ( frac{1}{x} ) is undefined.

Therefore, the equation is valid for all real numbers except ( x 0 ).

In conclusion, the original equation simplifies to an identity and is true for all real numbers ( x ) except when ( x 0 ).