solve the equation. check the solution.

a/a^2-16 + 2/a-4 = 2/a+4

a) -16,-4
b) 4
c) -16,4
d) -16

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : 

           (a)/(a^2-16)+(2/(a-4))-(2/(a+4))=0 

 3.1      Factoring:  a2 - 16 

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A2 - AB + BA - B2 =
         A2 - AB + AB - B2 = 
         A2 - B2

Note :  AB = BA is the commutative property of multiplication. 

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 16 is the square of 4
Check :  a2  is the square of  a1 

Factorization is :       (a + 4)  •  (a - 4) 

 4.1    Find the Least Common Multiple 

      The left denominator is :       (a+4) • (a-4) 

      The right denominator is :       a-4 

      Least Common Multiple: 
      (a+4) • (a-4) 

 4.2    Calculate multipliers for the two fractions 

    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 1

   Right_M = L.C.M / R_Deno = a+4

 4.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)2   and  (y2+y)/(y+1)3  are equivalent as well. 

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

 4.4       Adding up the two equivalent fractions 
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 5.1    Find the Least Common Multiple 

      The left denominator is :       (a+4) • (a-4) 

      The right denominator is :       a+4 

      Least Common Multiple: 
      (a+4) • (a-4) 

 5.2    Calculate multipliers for the two fractions 

    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 1

   Right_M = L.C.M / R_Deno = a-4

 5.3      Rewrite the two fractions into equivalent fractions

 5.4       Adding up the two equivalent fractions 

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

Now, on the left hand side, the  (a+4) • (a-4)  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   a+16  = 0

 6.2      Solve  :    a+16 = 0 

 Subtract  16  from both sides of the equation : 
                      a = -16 

                   a = -16

the 3rd option is right

step-by-step explanation: