3n/(n2 -7n + 10) - 2n/(n2-8n+15)
thanks
n^2-7n+10 = (n-2)(n-5)
n^2-8n+15 = (n-3)(n-5)
LCD = (n-2)(n-3)(n-5), so we have
[3n(n-3) - 2n(n-2)] / ((n-2)(n-3)(n-5))
= (3n^2-9n-2n^2+4n) / (...)
= (n^2-5n) / (...)
= n(n-5) / (n-2)(n-3)(n-5)
= n / (n-2)(n-3)
= 3/(n-3) - 2/(n-2)
To simplify the expression 3n/(n^2 - 7n + 10) - 2n/(n^2 - 8n + 15), we need to find a common denominator for both fractions and then combine them.
Step 1: Factorize the denominators.
The denominator of the first fraction, n^2 - 7n + 10, can be factored as (n - 5)(n - 2).
The denominator of the second fraction, n^2 - 8n + 15, can be factored as (n - 5)(n - 3).
Step 2: Find the common denominator.
To find the common denominator, we need to identify the highest power of each factor in the two denominators. In this case, the common denominator is (n - 5)(n - 2)(n - 3).
Step 3: Rewrite the fractions with the common denominator.
Rewriting the original expression with the common denominator:
3n/(n - 5)(n - 2) - 2n/(n - 5)(n - 3)
Step 4: Combine the fractions.
Now, we can subtract the two fractions since they have the same denominator:
[3n - 2n]/[(n - 5)(n - 2)(n - 3)]
Simplifying the numerator:
3n - 2n = n
Final simplified expression:
n/[(n - 5)(n - 2)(n - 3)]
Therefore, the simplified form of the expression 3n/(n^2 - 7n + 10) - 2n/(n^2 - 8n + 15) is n/[(n - 5)(n - 2)(n - 3)].