How do you combine #x / (x^2-4) + (3x - 5) /(x^2 + 4x + 4)#?

Question

How do you combine #x / (x^2-4)  +  (3x - 5) /(x^2 + 4x + 4)#?

Kennedy Adams · Accepted Answer

In order to create a common denominator note that
#(x^2-4)(x+2) = (x-2)(x+2)(x+2) = (x-2)(x^2+4x+4)# So
#x/(x^2-4) + (3x-5)/(x^2+4x+4)# #=(x(x+2) +(3x-5)(x-2))/((x-2)(x+2)(x+2))# #= (x^2+2x +3x^2-11x+15)/((x-2)(x+2)(x+2))# #=(4x^2-9x+15)/(x^3+2x^2-4x-8)#

Jace Anderson · Answer

undefined To combine the given expressions, we need to find a common denominator. The denominators are (x^2-4) and (x^2 + 4x + 4). The first denominator can be factored as (x-2)(x+2), and the second denominator can be factored as (x+2)(x+2).

The common denominator is (x-2)(x+2)(x+2).

Now, we can rewrite the fractions with the common denominator:

x / (x^2-4) = x / [(x-2)(x+2)]

(3x - 5) /(x^2 + 4x + 4) = (3x - 5) / [(x+2)(x+2)]

Combining the fractions, we add the numerators and keep the common denominator:

[x / [(x-2)(x+2)]] + [(3x - 5) / [(x+2)(x+2)]]

To add the fractions, we need to have the same denominator, which we already have.

Combine the numerators:

[x + (3x - 5)] / [(x-2)(x+2)(x+2)]

Simplify the numerator:

[4x - 5] / [(x-2)(x+2)(x+2)]

Therefore, the combined expression is (4x - 5) / [(x-2)(x+2)(x+2)].