Perform the indicated operations and reduce to lowest terms. Assume that no denominator has a value of zero.
$\frac{4-x^2}{x^2-4 x+4} \cdot \frac{7}{x+2}$
Answer (2)
Factor the numerator as a difference of squares: 4 − x 2 = − ( x − 2 ) ( x + 2 ) .
Factor the denominator as a perfect square: x 2 − 4 x + 4 = ( x − 2 ) 2 .
Rewrite the expression: ( x − 2 ) 2 − ( x − 2 ) ( x + 2 ) ⋅ x + 2 7 .
Cancel common factors to get the simplified expression: x − 2 − 7 .
Explanation
Initial Analysis We are asked to simplify the expression x 2 − 4 x + 4 4 − x 2 "." ⋅ x + 2 7 . To do this, we will factor the numerator and denominator of the first fraction and then cancel common factors.
Factoring the Numerator First, we factor the numerator 4 − x 2 as a difference of squares: 4 − x 2 = ( 2 − x ) ( 2 + x ) = − ( x − 2 ) ( x + 2 ) .
Factoring the Denominator Next, we factor the denominator x 2 − 4 x + 4 as a perfect square: x 2 − 4 x + 4 = ( x − 2 ) 2 = ( x − 2 ) ( x − 2 ) .
Rewriting the Expression Now we rewrite the original expression with the factored forms: x 2 − 4 x + 4 4 − x 2 ⋅ x + 2 7 = ( x − 2 ) ( x − 2 ) − ( x − 2 ) ( x + 2 ) ⋅ x + 2 7 .
Canceling Common Factors We can now cancel common factors. We have a factor of ( x − 2 ) in the numerator and denominator, and a factor of ( x + 2 ) in the numerator and denominator. Canceling these factors, we get: ( x − 2 ) ( x − 2 ) − ( x − 2 ) ( x + 2 ) ⋅ x + 2 7 = x − 2 − 1 ⋅ 1 7 = x − 2 − 7 .
Final Answer Thus, the simplified expression is x − 2 − 7 .
Examples
Simplifying rational expressions is a fundamental skill in algebra, which is useful in various real-world applications. For instance, when designing structures, engineers often encounter complex equations involving rational functions. Simplifying these expressions can make calculations easier and more manageable, ensuring the structural integrity and efficiency of the design. Similarly, in economics, simplifying rational expressions can aid in analyzing cost-benefit ratios and optimizing resource allocation.
To simplify the expression x 2 − 4 x + 4 4 − x 2 ⋅ x + 2 7 , we factor the numerator and denominator. After canceling common factors, the simplified expression is x − 2 − 7 .
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