Question

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

Answer 1

Answer:

The sum is $\frac{2d^2-5d+24}{d^2-3d-10}$

Step-by-step explanation:

The given expression is

$\frac{d^2-7d+12}{d^2-d-6}+\frac{d^2+5d+6}{d^2-2d-15}$

These are all quadratic trinomials, so we split each of the middle terms to obtain,

$\frac{d^2-4d-3d+12}{d^2-3d+2d-6}+\frac{d^2+2d+3d+6}{d^2-5d+3d-15}$

We factor to obtain,

$\frac{d(d-4)-3(d-4)}{d(d-3)+2(d-3)}+\frac{d(d+2)+3(d+2)}{d(d-5)+3(d-5)}$

We factor further to obtain,

$\frac{(d-4)(d-3)}{(d-3)(d+2)}+\frac{(d+2)(d+3)}{(d-5)(d+3)}$

We cancel out the common factors to get,

$\frac{(d-4)}{(d+2)}+\frac{(d+2)}{(d-5)}$

The least common denominator is $(d+2)(d-5)$.

We collect LCD to get

$\frac{(d-4)(d-5)+(d+2)(d+2)}{(d+2)(d-5)}$

We now expand to get,

$\frac{d^2-5d-4d+20+d^2+2d+2d+4}{d^2-5d+2d-10}$

We simplify to get,

$\frac{d^2-9d+20+d^2+4d+4}{d^2-3d-10}$

We simplify further to get,

$\frac{2d^2-5d+24}{d^2-3d-10}$