2 min read Jun 17, 2024

Exploring the Expression (x² + 1/x²) - 4(x + 1/x) + 6

This article delves into the algebraic expression (x² + 1/x²) - 4(x + 1/x) + 6, exploring its simplification, factorization, and potential applications.

Simplifying the Expression

The expression can be simplified by utilizing algebraic manipulation and recognizing patterns:

  1. Substitution: Let's introduce a new variable, y = x + 1/x. This substitution allows us to rewrite the expression in a more manageable form.
    • Notice that y² = (x + 1/x)² = x² + 2 + 1/x². Therefore, x² + 1/x² = y² - 2.
  2. Substitution and Simplification:
    • Substituting y and into the original expression gives us:
      (y² - 2) - 4y + 6 = y² - 4y + 4.

Factoring the Expression

The simplified expression, y² - 4y + 4, is a perfect square trinomial. It can be factored as:

(y - 2)²

Re-substitution and Solution

Finally, substituting back y = x + 1/x, we obtain the factored form:

(x + 1/x - 2)²

This expression represents a perfect square, always non-negative for any real value of x.


This expression, while seemingly abstract, can find applications in various areas, including:

  • Calculus: Finding derivatives and integrals of similar expressions.
  • Physics: Modeling certain physical phenomena that involve quadratic relationships.
  • Engineering: Analyzing and optimizing systems with similar mathematical representations.


Through simplification and factorization, we have transformed a complex expression into a more understandable and manageable form. This process reveals valuable insights into the nature of the expression and its potential applications across different fields.

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