Question
simplify the difference (a^2 - 2a - 3)/(a^2 - 9a + 18) - (a^2 - 5a - 6)/(a^2 + 9a + 8)
Answers
GPT 3.5
The given expression is (a^2 - 2a - 3)/(a^2 - 9a + 18) - (a^2 - 5a - 6)/(a^2 + 9a + 8).
First, factor the denominators of each fraction:
(a^2 - 9a + 18) = (a - 6)(a - 3)
(a^2 + 9a + 8) = (a + 1)(a + 8)
Now, rewrite the expression with the factored denominators:
(a^2 - 2a - 3)/((a - 6)(a - 3)) - (a^2 - 5a - 6)/((a + 1)(a + 8))
Now find a common denominator for both fractions, which is (a - 6)(a - 3)(a + 1)(a + 8):
[(a^2 - 2a - 3)(a + 1)(a + 8) - (a^2 - 5a - 6)(a - 6)(a - 3)] / [(a - 6)(a - 3)(a + 1)(a + 8)]
Then expand the numerator:
[(a^3 + 10a^2 + 17a + 24) - (a^3 - 11a^2 + 36a - 18)] / [(a - 6)(a - 3)(a + 1)(a + 8)]
Simplify the numerator:
(21a^2 + 19a + 42) / [(a - 6)(a - 3)(a + 1)(a + 8)]
Therefore, the simplified expression is (21a^2 + 19a + 42) / [(a - 6)(a - 3)(a + 1)(a + 8)].
First, factor the denominators of each fraction:
(a^2 - 9a + 18) = (a - 6)(a - 3)
(a^2 + 9a + 8) = (a + 1)(a + 8)
Now, rewrite the expression with the factored denominators:
(a^2 - 2a - 3)/((a - 6)(a - 3)) - (a^2 - 5a - 6)/((a + 1)(a + 8))
Now find a common denominator for both fractions, which is (a - 6)(a - 3)(a + 1)(a + 8):
[(a^2 - 2a - 3)(a + 1)(a + 8) - (a^2 - 5a - 6)(a - 6)(a - 3)] / [(a - 6)(a - 3)(a + 1)(a + 8)]
Then expand the numerator:
[(a^3 + 10a^2 + 17a + 24) - (a^3 - 11a^2 + 36a - 18)] / [(a - 6)(a - 3)(a + 1)(a + 8)]
Simplify the numerator:
(21a^2 + 19a + 42) / [(a - 6)(a - 3)(a + 1)(a + 8)]
Therefore, the simplified expression is (21a^2 + 19a + 42) / [(a - 6)(a - 3)(a + 1)(a + 8)].