Which of the following is a simplification of
\[\frac{2 x^2-9 x-18}{x^2-6 x}-\frac{3 x^2+8 x-3}{x^2+3 x}\]
A. \[\frac{4}{x}-1\]
B. \[\frac{2}{x}-1\]
C. \[2-\frac{1}{x}\]
D. \[\frac{4}{x}-2\]
E. \[4-x\]
Answer (1)
Factor the numerators and denominators of both rational expressions.
Cancel out the common factors in each fraction.
Combine the two fractions into a single fraction.
Simplify the resulting expression to get the final answer: x 4 − 1 .
Explanation
Problem Analysis We are given the expression x 2 − 6 x 2 x 2 − 9 x − 18 − x 2 + 3 x 3 x 2 + 8 x − 3 . Our goal is to simplify this expression and determine which of the provided options is equivalent to it.
Factoring Expressions First, we factor the quadratic expressions in the numerators and the expressions in the denominators.
The numerator of the first fraction is 2 x 2 − 9 x − 18 . We are looking for two numbers that multiply to 2 × − 18 = − 36 and add up to − 9 . These numbers are − 12 and 3 . Thus, we can write 2 x 2 − 9 x − 18 = 2 x 2 − 12 x + 3 x − 18 = 2 x ( x − 6 ) + 3 ( x − 6 ) = ( 2 x + 3 ) ( x − 6 ) .
The denominator of the first fraction is x 2 − 6 x = x ( x − 6 ) .
The numerator of the second fraction is 3 x 2 + 8 x − 3 . We are looking for two numbers that multiply to 3 × − 3 = − 9 and add up to 8 . These numbers are 9 and − 1 . Thus, we can write 3 x 2 + 8 x − 3 = 3 x 2 + 9 x − x − 3 = 3 x ( x + 3 ) − 1 ( x + 3 ) = ( 3 x − 1 ) ( x + 3 ) .
The denominator of the second fraction is x 2 + 3 x = x ( x + 3 ) .
Rewriting with Factored Forms Now we rewrite the expression with the factored forms: x ( x − 6 ) ( 2 x + 3 ) ( x − 6 ) − x ( x + 3 ) ( 3 x − 1 ) ( x + 3 )
Canceling Common Factors We cancel the common factors in each fraction: x 2 x + 3 − x 3 x − 1
Combining Fractions Combine the fractions: x ( 2 x + 3 ) − ( 3 x − 1 )
Simplifying the Numerator Simplify the numerator: x 2 x + 3 − 3 x + 1 = x − x + 4 = x 4 − x
Rewriting the Expression Rewrite the expression as: x 4 − x x = x 4 − 1
Final Answer Comparing the simplified expression x 4 − 1 with the given choices, we see that it matches option A.
Examples
Simplifying rational expressions is a fundamental skill in algebra, useful in various real-world applications. For instance, when designing a bridge, engineers use rational functions to model the load distribution and structural stress. Simplifying these expressions allows them to efficiently calculate critical parameters, ensuring the bridge's stability and safety. Similarly, in electrical engineering, simplifying complex circuit equations often involves rational expressions, aiding in the design and analysis of efficient circuits.
