3n/(n^2 -7n + 10) - 2n/(n^2-8n+15)
Factor out n/(n-5) from both terms
To simplify the given expression (3n/(n^2 -7n + 10) - 2n/(n^2-8n+15)), we first need to factor the denominators and find a common denominator.
Let's start by factoring the denominators:
For the first denominator, n^2 -7n + 10, we need to find two numbers that multiply to give 10 and add up to -7. The numbers are -5 and -2:
n^2 -7n + 10 = (n - 5)(n - 2)
For the second denominator, n^2 -8n+15, we need to find two numbers that multiply to give 15 and add up to -8. The numbers are -5 and -3:
n^2 -8n + 15 = (n - 5)(n - 3)
Now that we have factored the denominators, we can rewrite the expression using the factored forms:
3n/(n - 5)(n - 2) - 2n/(n - 5)(n - 3)
Next, we need to find a common denominator. In this case, since both denominators have the same factors, (n - 5) and (n - 2), we can directly use their product as the common denominator:
Common denominator = (n - 5)(n - 2)
Now, we can rewrite the expression with the common denominator:
(3n(n - 3))/(n - 5)(n - 2) - (2n(n - 2))/(n - 5)(n - 2)
Simplifying further:
(3n^2 - 9n - 2n^2 + 4n)/(n - 5)(n - 2)
Combining like terms:
(n^2 - 5n)/(n - 5)(n - 2)
Hence, the simplified form of the expression 3n/(n^2 -7n + 10) - 2n/(n^2-8n+15) is (n^2 - 5n)/(n - 5)(n - 2).