Difficulty: Hard

Average Score: 33%

Which of the following expressions is equivalent to x/(x ^ 2 + 6x + 9) - (x - 3)/(x ^ 2 + 5x + 6) ?

Rationale

(2x - 9)/((x + 2)(x + 3))

To simplify the expression x/(x^2 + 6x + 9) - (x - 3)/(x^2 + 5x + 6), we first factor the denominators, resulting in x/((x + 3)^2) - (x - 3)/((x + 2)(x + 3)). After finding a common denominator and simplifying, the final expression equates to (2x - 9)/((x + 2)(x + 3)).

A ((2x - 9)/((x + 2)(x + 3))) CORRECT

This expression represents the correct simplification of the original problem. By combining the fractions over a common denominator and simplifying, we arrive at this exact form, confirming its equivalence to the original expression.

B ((2x + 9)/((x + 2)(x + 3))) YOUR ANSWER

This option introduces an incorrect sign in the numerator. The correct simplification yields (2x - 9), not (2x + 9), indicating a fundamental error in the algebraic manipulation of the original expression.

C ((2x - 9)/((x + 2) * (x + 3) ^ 2)) YOUR ANSWER

This choice incorrectly squares the factor (x + 3) in the denominator. The correct common denominator is simply (x + 2)(x + 3), as the correct factors of the original denominators do not indicate a squared term.

D ((2x + 9)/((x + 2) * (x + 3) ^ 2)) YOUR ANSWER

Similar to choice C, this option incorrectly combines a sign error in the numerator with a squared term in the denominator. Neither the sign nor the factorization matches the result of the simplification process.

E ((2x + 9)/((x + 2) ^ 2 * (x + 3) ^ 2)) YOUR ANSWER

This expression is incorrect as it also features both sign and factorization errors. The presence of squared terms for both factors in the denominator does not originate from the original expressions, leading to its inaccuracy.

Conclusion

The correct simplification of the given expression is (2x - 9)/((x + 2)(x + 3)), as derived from the proper factoring and combining of the original fractions. All other choices introduce errors in either the numerator or denominator, demonstrating a misunderstanding of the algebraic principles involved in simplification. Thus, choice A remains the only valid equivalent expression.

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