[tex]x^2+\frac{9}{x^2}-6[/tex]

Rewrite the expression as a perfect square and simplify: x 2 + x 2 9 ​ − 6 = ( x − x 3 ​ ) 2 . The simplified expression is ( x − x 3 ​ ) 2 ​ .
Explanation

Understanding the Expression We are given the expression x 2 + x 2 9 ​ − 6 and we want to simplify it.

Rewriting the Expression We can rewrite the expression as x 2 − 6 + x 2 9 ​ . Notice that this looks similar to the expansion of a squared binomial.

Recognizing a Perfect Square Recall that ( a − b ) 2 = a 2 − 2 ab + b 2 . Let's see if we can rewrite our expression in this form. We have x 2 which can be a 2 , and x 2 9 ​ which can be b 2 . If a = x and b = x 3 ​ , then ( x − x 3 ​ ) 2 = x 2 − 2 ( x ) ( x 3 ​ ) + ( x 3 ​ ) 2 = x 2 − 6 + x 2 9 ​ .

Final Simplification Therefore, the simplified expression is ( x − x 3 ​ ) 2 .


Examples
In physics, this type of expression can appear when analyzing the potential energy of a system. For example, if you have a particle whose potential energy U ( x ) is given by U ( x ) = x 2 + x 2 9 ​ − 6 , you can simplify this to U ( x ) = ( x − x 3 ​ ) 2 . This form might be easier to analyze to find equilibrium points or understand the stability of the system.

Answered by GinnyAnswer | 2025-07-08