Can you please help me solve this equation

Question

(x+11)/(x+4)+(56)/(x²-16)=0

Answer 1

(x+11)/(x+4)+(56)/(x²-16)=0

(x+11)/(x+4)+(56)/(x+4)(x-4)=0

(x+11)(x-4)/(x+4)(x-4)+(56)/(x+4)(x-4)=0

[(x+11)(x-4)+56]/(x+4)(x-4)=0

(x+11)(x-4)+56=0

x²+11x-4x-44+56=0

x²+7x+12=0

(x+4)(x+3)=0

x+4=0

x=-4 this is not a solution because it is not in the domain of x because -4 makes both fractions have a 0 in the denominator and that is undefined

x+3=0

x=-3 is the solution

 

Answer 2

In order to add the two fractions, we need a common denominator.  Notice that (x²-16) is the difference of squares, so it is (x+4)(x-4).

That’s convenient, since the denominator of the first fraction is (x+4). 

Let’s multiply everything by (x+4)(x-4), that is, (x²-16):

 

  (x+11)/(x+4)+(56)/(x2-16)=0

   (x+11)(x-4)  + 56 = 0

    x²+ 7x -44   + 56 = 0       [use FOIL]

    x²+ 7x + 12 = 0

    (x+3)(x+4) = 0                [either factor or use quadratic formula]

Either  x=-3  or x=-4

x=-4 is not in the domain (the allowable values of x) because it makes the

    fractions undefined in the problem statement

x =-3is the solution

 

Checking (very important):

  Is (-3+11)/(-3+4)  +  (56)/((-3)²-16) = 0     ?

       8/1          + 56/(-7)   = 0   ?

       8    -  8   =  0   ? yes