If x ≠ 0 and x ≠ 3, then (3/(2x)^2)- (2/(x² - 3x)=

Rationale

-x - 9/2x^3 - 6x².

The expression simplifies to -x - 9/2x^3 - 6x² after performing algebraic operations, including finding a common denominator and simplifying the resulting fractions. Each step adheres to the rules of algebraic manipulation, confirming the correctness of this expression.

A (1/x² - 3x) YOUR ANSWER

This option does not correctly represent the result of the original expression. The term 1/x² does not appear in the simplification process, and the subtraction of 3x does not align with the derived terms from the algebraic manipulation.

B (3 - 4x/2x² - 3x) YOUR ANSWER

This expression incorrectly combines terms and does not reflect the simplification of the original equation. The presence of the 3 term and the incorrect arrangement of the x components lead to a misrepresentation of the final expression.

C (-x - 9/2x^3 - 6x²) CORRECT

This choice accurately captures the result obtained from the algebraic simplification of the original expression. Each component aligns with the necessary steps to reduce the expression correctly, confirming its validity.

D (7x - 9/2x^3 - 6x²) YOUR ANSWER

This option introduces an incorrect 7x term that does not derive from the original expression. The alteration of coefficients does not align with the algebraic processes needed to simplify the given equation, making it an incorrect choice.

Conclusion

The expression simplifies to -x - 9/2x^3 - 6x², highlighting essential algebraic principles in manipulating rational expressions. Each incorrect choice fails to adhere to these principles, either misrepresenting terms or altering coefficients, which confirms that option C is the only correct simplification. Understanding these transformations is crucial for effectively solving similar algebraic problems.

If x ≠ 0 and x ≠ 3, then (3/(2x)^2)- (2/(x² - 3x)=

-x - 9/2x^3 - 6x².

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