Simplify the sum. d^2-7d+12/d^2-d-6 + d^2+5d+6/d^2-2d-15

Answer 1

To simplify the sum, we need to factor and simplify each rational expression separately before adding them together.

Given expressions:
1. (d^2-7d+12)/(d^2-d-6)
2. (d^2+5d+6)/(d^2-2d-15)

1. Factoring the first rational expression:
(d^2-7d+12) = (d-3)(d-4)
(d^2-d-6) = (d-3)(d+2)

So, the first rational expression simplifies to:
(d-3)(d-4)/(d-3)(d+2)
(d-4)/(d+2)

2. Factoring the second rational expression:
(d^2+5d+6) = (d+2)(d+3)
(d^2-2d-15) = (d-5)(d+3)

So, the second rational expression simplifies to:
(d+2)(d+3)/(d-5)(d+3)
(d+2)/(d-5)

Now, the simplified sum of the two rational expressions is:
(d-4)/(d+2) + (d+2)/(d-5)

To add the two fractions, we need to find a common denominator, which is (d+2)(d-5). So, we have:
((d-4)(d-5) + (d+2)(d+2))/((d+2)(d-5))

Expanding the numerators and simplifying, we get:
(d^2 - 9d + 20 + d^2 + 4d + 4)/(d^2 - 3d - 10)
(2d^2 - 5d + 24)/(d^2 - 3d - 10)

Therefore, the simplified sum of the given expressions is:
(2d^2 - 5d + 24)/(d^2 - 3d - 10)