\frac{2}{x^2-16}-\frac{1}{x^2-4 x}
Answer (2)
Factor the denominators: x 2 − 16 = ( x − 4 ) ( x + 4 ) and x 2 − 4 x = x ( x − 4 ) .
Find the least common denominator (LCD): x ( x − 4 ) ( x + 4 ) .
Rewrite each fraction with the LCD and combine them: x ( x − 4 ) ( x + 4 ) 2 x − ( x + 4 ) .
Simplify the expression by canceling the common factor ( x − 4 ) : x ( x + 4 ) 1 .
Explanation
Understanding the Problem We are asked to simplify the expression x 2 − 16 2 − x 2 − 4 x 1 . This involves rational functions, and we need to find a common denominator and combine the fractions. Factoring the denominators will help us find the least common denominator.
Factoring the Denominators First, let's factor the denominators:
x 2 − 16 = ( x − 4 ) ( x + 4 ) and x 2 − 4 x = x ( x − 4 )
Finding the Least Common Denominator Now, let's find the least common denominator (LCD). The LCD is x ( x − 4 ) ( x + 4 ) .
Rewriting Fractions with the LCD Next, we rewrite each fraction with the LCD:
( x − 4 ) ( x + 4 ) 2 = x ( x − 4 ) ( x + 4 ) 2 x and x ( x − 4 ) 1 = x ( x − 4 ) ( x + 4 ) x + 4
Combining the Fractions Now, we combine the fractions:
x ( x − 4 ) ( x + 4 ) 2 x − x ( x − 4 ) ( x + 4 ) x + 4 = x ( x − 4 ) ( x + 4 ) 2 x − ( x + 4 )
Simplifying the Numerator Simplify the numerator:
2 x − ( x + 4 ) = 2 x − x − 4 = x − 4
Simplifying the Expression Simplify the expression:
x ( x − 4 ) ( x + 4 ) x − 4
Canceling Common Factors Cancel the common factor ( x − 4 ) from the numerator and denominator:
x ( x − 4 ) ( x + 4 ) x − 4 = x ( x + 4 ) 1
Final Answer The final simplified expression is x ( x + 4 ) 1 .
Examples
Rational expressions are useful in various fields, such as physics and engineering, where they can model complex relationships between variables. For example, in electrical engineering, rational functions can describe the impedance of a circuit as a function of frequency. Simplifying these expressions allows engineers to analyze and design circuits more efficiently. Similarly, in physics, rational functions can appear in equations describing the motion of objects or the behavior of waves. By simplifying these expressions, physicists can gain a better understanding of the underlying phenomena and make more accurate predictions. Understanding how to manipulate and simplify rational expressions is a fundamental skill that enables professionals to solve real-world problems in these fields.
To simplify the expression x 2 − 16 2 − x 2 − 4 x 1 , we factor the denominators, find the least common denominator, rewrite each fraction, and combine them. After simplifying the numerator and canceling common factors, the final expression is x ( x + 4 ) 1 .
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