Please Simplify the difference.
open parentheses a squared minus two a minus three over a squared minus nine a plus eighteen closed parentheses minus a squared minus five a minus six over a squared plus a plus eight
The simplified difference between the two fractions is (a - 3) / (a - 3)(a - 6)
Are you sure?
I apologize for the incorrect simplification. Here is the correct simplified difference:
(a - 1) / (a - 3)(a - 6)
I don't think that is correct.
I apologize for the confusion earlier. Let's simplify the original expression step by step:
Given expression: (a^2 - 2a - 3) / (a^2 - 9a + 18) - (a^2 - 5a - 6) / (a^2 + a + 8)
Factor the denominators:
(a^2 - 2a - 3) = (a - 3)(a + 1)
(a^2 - 9a + 18) = (a - 6)(a - 3)
(a^2 - 5a - 6) = (a - 6)(a + 1)
(a^2 + a + 8) does not factor further.
Now substitute the factored denominators back into the expression:
[(a - 3)(a + 1)] / [(a - 6)(a - 3)] - [(a - 6)(a + 1)] / (a^2 + a + 8)
Cancel out common factors:
= (a + 1) / (a - 6) - (a + 1) / (a^2 + a + 8)
Common denominator:
= [(a + 1)(a^2 + a + 8) - (a + 1)(a - 6)] / (a^2 + a + 8)(a - 6)
Expand and simplify:
= (a^3 + 2a^2 + 8a + a^2 + a + 8 - a^2 - 6a - a - 6) / (a^3 - 6a^2 + 8a^2 - 48a + 8a - 48)
= (a^3 + 3a^2 + 2) / (a^3 + 2a^2 - 40a - 48)
Therefore, the simplified difference between the two fractions is (a^3 + 3a^2 + 2) / (a^3 + 2a^2 - 40a - 48).
My assignment doesn't have this as an available option. Is there another answer?
Apologies for the confusion earlier. Let's go over the simplification again:
Given expression: (a^2 - 2a - 3) / (a^2 - 9a + 18) - (a^2 - 5a - 6) / (a^2 + a + 8)
Factor the denominators:
(a^2 - 2a - 3) = (a - 3)(a + 1)
(a^2 - 9a + 18) = (a - 6)(a - 3)
(a^2 - 5a - 6) = (a - 6)(a + 1)
(a^2 + a + 8) does not factor further.
Now substitute the factored denominators back into the expression and find a common denominator:
= [(a - 3)(a + 1) / (a - 6)(a - 3) - (a - 6)(a + 1) / (a^2 + a + 8)]
Combine the fractions over the common denominator:
= [(a - 3)(a + 1) - (a - 6)(a + 1)] / [((a - 6)(a - 3))]
Expand and simplify the numerator:
= [a^2 - 2a - 3 - (a^2 - 5a - 6)] / [(a - 6)(a - 3)]
= (a^2 - 2a - 3 - a^2 + 5a + 6) / (a - 6)(a - 3)
= (3a + 3) / (a - 6)(a - 3)
Therefore, the simplified difference between the two fractions is (3a + 3) / (a - 6)(a - 3).
is that the same as
negative three a plus forty-four over open parentheses a minus six closed parentheses open parentheses a plus eight closed parentheses?
To check if (3a + 3) / ((a - 6)(a - 3)) simplifies to the same expression as -3a + 44 / (a - 6)(a + 8), we need to adjust the terms in the first expression:
(3a + 3) / ((a - 6)(a - 3))
Here we can simplify the numerator by factoring out a common factor of 3:
3(a + 1) / ((a - 6)(a - 3))
This simplifies to:
3(a + 1) / ((a - 6)(a - 3)) = 3(a + 1) / (a - 6)(a - 3)
Therefore, the simplified expression for (3a + 3) / ((a - 6)(a - 3)) is not equivalent to -3a + 44 / (a - 6)(a + 8).