-4(x%2B1%2Fx)%2B6.jpeg "(x^2+1/x^2)-4(x+1/x)+6")
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:
- 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.
- Substitution and Simplification:
- Substituting y and y² into the original expression gives us:
(y² - 2) - 4y + 6 = y² - 4y + 4.
- Substituting y and y² into the original expression gives us:
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.
Applications
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.
Conclusion
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.