2/x2-9 + 1/3x+9
What is the question?
that right there is the question
To simplify the expression (2/x^2 - 9) + (1/3x + 9), we need to find a common denominator and combine the fractions.
Step 1: First, let's simplify each fraction individually.
The fraction 2/x^2 - 9 can be left as it is since there is no way to simplify it further.
The fraction 1/3x + 9 cannot be simplified any further since the terms in the denominator (3x and 9) do not share any common factors.
Step 2: Now, we need to find a common denominator for the two fractions. The first fraction has a denominator of x^2 - 9, and the second fraction has a denominator of 3x.
To find the common denominator, we need to factor the expression x^2 - 9 and check if any factors are missing from the other fraction's denominator.
The expression x^2 - 9 can be factored as (x + 3)(x - 3). So, the common denominator is (x + 3)(x - 3)(3x).
Step 3: Adjust the fractions to have the common denominator.
For the first fraction, multiply the numerator and denominator by (3x) to have a common denominator:
2/x^2 - 9 = 2(3x) / [(x + 3)(x - 3)(3x)]
For the second fraction, multiply the numerator and denominator by (x + 3)(x - 3) to have a common denominator:
1/3x + 9 = [(x + 3)(x - 3)] / [(x + 3)(x - 3)(3x)]
Step 4: Combine the fractions by adding their numerators.
(2(3x) / [(x + 3)(x - 3)(3x)]) + ([(x + 3)(x - 3)] / [(x + 3)(x - 3)(3x)])
= (6x / [(x + 3)(x - 3)(3x)]) + ([(x + 3)(x - 3)] / [(x + 3)(x - 3)(3x)])
Step 5: Simplify the expression.
Since both fractions have the same denominator, we can simply add their numerators:
= (6x + (x + 3)(x - 3)) / [(x + 3)(x - 3)(3x)]
= (6x + (x^2 - 9)) / [(x + 3)(x - 3)(3x)]
= (x^2 + 6x - 9) / [(x + 3)(x - 3)(3x)]
Therefore, the simplified expression is (x^2 + 6x - 9) / [(x + 3)(x - 3)(3x)].