Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "a2"   was replaced by   "a^2". 

Step  1  :

            3
 Simplify   —
            4

Equation at the end of step  1  :

       1         3             1      3
  (((((—•(a2))-((—•a)•b))+ac)+(—•a))-(—•b))+c
       2         4             2      4

Step  2  :

            1
 Simplify   —
            2

Equation at the end of step  2  :

       1         3             1     3b
  (((((—•(a2))-((—•a)•b))+ac)+(—•a))-——)+c
       2         4             2     4 

Step  3  :

            3
 Simplify   —
            4

Equation at the end of step  3  :

       1         3            a  3b
  (((((—•(a2))-((—•a)•b))+ac)+—)-——)+c
       2         4            2  4 
 Step  4  :
            1
 Simplify   —
            2

Equation at the end of step  4  :

       1     3ab      a  3b
  (((((—•a2)-———)+ac)+—)-——)+c
       2      4       2  4 

Step  5  :

Equation at the end of step  5  :

      a2    3ab            a     3b     
  ((((—— -  ———) +  ac) +  —) -  ——) +  c
      2      4             2     4      

Step  6  :

Calculating the Least Common Multiple :

 6.1    Find the Least Common Multiple

      The left denominator is :       2 

      The right denominator is :       4 

      Least Common Multiple:
      4 

Calculating Multipliers :

 6.2    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 = 2

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 6.3      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.

   L. Mult. • L. Num.      a2 • 2
   ——————————————————  =   ——————
         L.C.M               4   

   R. Mult. • R. Num.      3ab
   ——————————————————  =   ———
         L.C.M              4 

Adding fractions that have a common denominator :

 6.4       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:

 a2 • 2 - (3ab)     2a2 - 3ab
 ——————————————  =  —————————
       4                4    

Equation at the end of step  6  :

     (2a2 - 3ab)           a     3b     
  (((——————————— +  ac) +  —) -  ——) +  c
          4                2     4      

Step  7  :

Rewriting the whole as an Equivalent Fraction :

 7.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  4  as the denominator :

          ac     ac • 4
    ac =  ——  =  ——————
          1        4   

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   2a² - 3ab  =   a • (2a - 3b) 

Adding fractions that have a common denominator :

 8.2       Adding up the two equivalent fractions

 a • (2a-3b) + ac • 4     2a2 - 3ab + 4ac
 ————————————————————  =  ———————————————
          4                      4       

Equation at the end of step  8  :

    (2a2 - 3ab + 4ac)    a     3b     
  ((————————————————— +  —) -  ——) +  c
            4            2     4      

Step  9  :

Step  10  :

Pulling out like terms :

 10.1     Pull out like factors :

   2a² - 3ab + 4ac  =   a • (2a - 3b + 4c) 

Calculating the Least Common Multiple :

 10.2    Find the Least Common Multiple

      The left denominator is :       4 

      The right denominator is :       2 

      Least Common Multiple:
      4 

Calculating Multipliers :

 10.3    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 = 2

Making Equivalent Fractions :

 10.4      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      a • (2a-3b+4c)
   ——————————————————  =   ——————————————
         L.C.M                   4       

   R. Mult. • R. Num.      a • 2
   ——————————————————  =   —————
         L.C.M               4  

Adding fractions that have a common denominator :

 10.5       Adding up the two equivalent fractions

 a • (2a-3b+4c) + a • 2     2a2 - 3ab + 4ac + 2a
 ——————————————————————  =  ————————————————————
           4                         4          

Equation at the end of step  10  :

   (2a2 - 3ab + 4ac + 2a)    3b     
  (—————————————————————— -  ——) +  c
             4               4      

Step  11  :

Step  12  :

Pulling out like terms :

 12.1     Pull out like factors :

   2a² - 3ab + 4ac + 2a  = 

  a • (2a - 3b + 4c + 2) 

Trying to factor a multi variable polynomial :

 12.2       Split       2a - 3b + 4c + 2 
             into two 2-term polynomials
              - 3b + 2a   and     + 4c + 2
             This partition did not result in a factorization. We'll try another one:

             2a - 3b   and     + 4c + 2
             This partition did not result in a factorization. We'll try another one:

             2a + 4c   and     - 3b + 2
             This partition did not result in a factorization. We'll try another one:

             2a + 2   and     + 4c - 3b
             This partition did not result in a factorization. We'll try another one:

              + 2 + 2a   and     + 4c - 3b
             This partition did not result in a factorization. We'll try another one:

              + 4c + 2a   and     - 3b + 2
            All three partitions failed. Tiger finds no factorization

Adding fractions which have a common denominator :

 12.3       Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 a • (2a-3b+4c+2) - (3b)     2a2 - 3ab + 4ac + 2a - 3b
 ———————————————————————  =  —————————————————————————
            4                            4            

Equation at the end of step  12  :

  (2a2 - 3ab + 4ac + 2a - 3b)    
  ——————————————————————————— +  c
               4                 

Step  13  :

Rewriting the whole as an Equivalent Fraction :

 13.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  +4  as the denominator :

         c     c • 4
    c =  —  =  —————
         1       4  

Adding fractions that have a common denominator :

 13.2       Adding up the two equivalent fractions

 (2a2-3ab+4ac+2a-3b) + c • 4     2a2 - 3ab + 4ac + 2a - 3b + 4c
 ———————————————————————————  =  ——————————————————————————————
              4                                4               

Trying to factor by pulling out :

 13.3      Factoring:   + 2a² - 3ab + 4ac + 2a - 3b + 4c 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  -3ab - 3b 
Group 2:   + 4ac + 4c 
Group 3:   + 2a² + 2a 

Pull out from each group separately :

Group 1:   ( + a + 1) • (-3b)
Group 2:   ( + a + 1) • (4c)
Group 3:   ( + a + 1) • (2a)




   Add 1+2+3 : ————————————————
               ( + a + 1)  •  ( + 2a - 3b + 4c) 
Which is the desired factorization

Final result :

  (2a + 3b + 4c) • (a + 1)
  ————————————————————————
             4