What is the sum? [tex]$\frac{3}{x^2-9}+\frac{5}{x+3}$[/tex]

Factor the denominators of the given rational expressions.
Find a common denominator for all the expressions.
Add the numerators of the expressions.
Simplify the resulting expression by combining like terms and canceling common factors: x 2 − 9 5 x 2 + 8 x − 48 ​ ​

Explanation

Problem Setup We are asked to find the sum of the following expressions:

x 2 − 9 3 ​ + x + 3 5 ​ + x 2 + x − 6 8 ​ + x − 3 5 x − 12 ​ + ( x + 3 ) ( x − 3 ) − 5 x ​ + ( x + 3 ) ( x − 3 ) 5 x − 12 ​

Factoring Denominators First, let's factor the denominators where possible to make it easier to find a common denominator.

x 2 − 9 = ( x − 3 ) ( x + 3 ) x 2 + x − 6 = ( x + 3 ) ( x − 2 )

Rewriting with Factored Denominators Now we can rewrite the expression as:

( x − 3 ) ( x + 3 ) 3 ​ + x + 3 5 ​ + ( x + 3 ) ( x − 2 ) 8 ​ + x − 3 5 x − 12 ​ + ( x + 3 ) ( x − 3 ) − 5 x ​ + ( x + 3 ) ( x − 3 ) 5 x − 12 ​

Finding Common Denominator The common denominator for all these fractions is ( x − 3 ) ( x + 3 ) ( x − 2 ) . Now we rewrite each fraction with this common denominator:

( x − 3 ) ( x + 3 ) ( x − 2 ) 3 ( x − 2 ) ​ + ( x + 3 ) ( x − 3 ) ( x − 2 ) 5 ( x − 3 ) ( x − 2 ) ​ + ( x + 3 ) ( x − 2 ) ( x − 3 ) 8 ( x − 3 ) ​ + ( x − 3 ) ( x + 3 ) ( x − 2 ) ( 5 x − 12 ) ( x + 3 ) ( x − 2 ) ​ + ( x + 3 ) ( x − 3 ) ( x − 2 ) − 5 x ( x − 2 ) ​ + ( x + 3 ) ( x − 3 ) ( x − 2 ) ( 5 x − 12 ) ( x − 2 ) ​

Adding Numerators Now we add the numerators:

( x − 3 ) ( x + 3 ) ( x − 2 ) 3 ( x − 2 ) + 5 ( x − 3 ) ( x − 2 ) + 8 ( x − 3 ) + ( 5 x − 12 ) ( x + 3 ) ( x − 2 ) − 5 x ( x − 2 ) + ( 5 x − 12 ) ( x − 2 ) ​

Expanding and Simplifying Expanding and simplifying the numerator:

3 x − 6 + 5 ( x 2 − 5 x + 6 ) + 8 x − 24 + ( 5 x − 12 ) ( x 2 + x − 6 ) − 5 x 2 + 10 x + 5 x 2 − 10 x − 12 x + 24 = 3 x − 6 + 5 x 2 − 25 x + 30 + 8 x − 24 + 5 x 3 + 5 x 2 − 30 x − 12 x 2 − 12 x + 72 − 5 x 2 + 10 x + 5 x 2 − 22 x + 24 = 5 x 3 − 2 x 2 − 68 x + 96

Combining Terms So the expression becomes:

( x − 3 ) ( x + 3 ) ( x − 2 ) 5 x 3 − 2 x 2 − 68 x + 96 ​

Factoring Numerator and Denominator We can factor the denominator as ( x − 3 ) ( x + 3 ) ( x − 2 ) .
After factoring the numerator, we have: 5 x 3 − 2 x 2 − 68 x + 96 = ( x − 2 ) ( 5 x 2 + 8 x − 48 ) .
So the expression is: ( x − 3 ) ( x + 3 ) ( x − 2 ) ( x − 2 ) ( 5 x 2 + 8 x − 48 ) ​

Simplifying the Expression We can cancel the ( x − 2 ) term from the numerator and denominator:


( x − 3 ) ( x + 3 ) 5 x 2 + 8 x − 48 ​
However, 5 x 2 + 8 x − 48 cannot be factored easily to cancel out any more terms. Therefore, the simplified expression is:
x 2 − 9 5 x 2 + 8 x − 48 ​

Final Answer The sum of the given expressions is:

x 2 − 9 5 x 2 + 8 x − 48 ​
Examples
Understanding how to simplify complex algebraic expressions is crucial in many fields, such as physics and engineering. For example, when analyzing the motion of a projectile, you might encounter complex rational functions that describe the projectile's trajectory. Simplifying these functions, as we did in this problem, allows engineers to more easily predict the projectile's range, maximum height, and impact point. This ensures accuracy and efficiency in designing systems ranging from artillery to sports equipment.

Answered by GinnyAnswer | 2025-07-03

The sum of the expressions x 2 − 9 3 ​ + x + 3 5 ​ is x 2 − 9 5 x − 12 ​ . To combine the fractions, we found a common denominator and simplified the result. Thus, the answer shows how you can add rational expressions effectively.
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Answered by Anonymous | 2025-07-04