Simplify: [tex]$\frac{x}{x^2-9}+\frac{3}{9-x^2}$[/tex]
Answer (1)
Rewrite the second term with a common denominator: 9 − x 2 3 = − x 2 − 9 3 .
Combine the fractions: x 2 − 9 x − x 2 − 9 3 = x 2 − 9 x − 3 .
Factor the denominator: ( x − 3 ) ( x + 3 ) x − 3 .
Cancel the common factor: x + 3 1 . The simplified expression is x + 3 1 .
Explanation
Understanding the Problem We are asked to simplify the expression x 2 − 9 x + 9 − x 2 3 . The key here is to recognize that the denominators are very similar and can be made identical with a small manipulation.
Rewriting the Expression Notice that 9 − x 2 = − ( x 2 − 9 ) . We can rewrite the second term as follows: 9 − x 2 3 = − ( x 2 − 9 ) 3 = − x 2 − 9 3 Now our expression becomes: x 2 − 9 x − x 2 − 9 3
Combining Fractions Since the denominators are now the same, we can combine the fractions: x 2 − 9 x − x 2 − 9 3 = x 2 − 9 x − 3
Factoring the Denominator Now we factor the denominator. We recognize that x 2 − 9 is a difference of squares, so we can factor it as ( x − 3 ) ( x + 3 ) . Thus, our expression becomes: x 2 − 9 x − 3 = ( x − 3 ) ( x + 3 ) x − 3
Canceling Common Factors We can cancel the common factor of ( x − 3 ) from the numerator and the denominator, provided that x = 3 :
( x − 3 ) ( x + 3 ) x − 3 = x + 3 1 So, the simplified expression is x + 3 1 .
Final Answer Therefore, the simplified form of the given expression is x + 3 1 .
Examples
Simplifying rational expressions is a fundamental skill in algebra and is used extensively in calculus and other advanced mathematics courses. For example, when solving for the area under a curve, you might encounter an integral that involves rational functions. Simplifying these functions makes the integration process much easier. Also, in physics, simplifying complex equations involving rational expressions can help in modeling physical phenomena more efficiently.
