Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "23.8" was replaced by "(238/10)". 2 more similar replacement(s)

Step  1  :

            119
 Simplify   ———
             5 

Equation at the end of step  1  :

            103          119
  ((x2) -  (——— • x)) +  ———
            10            5 

Step  2  :

            103
 Simplify   ———
            10 

Equation at the end of step  2  :

            103          119
  ((x2) -  (——— • x)) +  ———
            10            5 
 Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  10  as the denominator :

           x2     x2 • 10
     x2 =  ——  =  ———————
           1        10   

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

Adding fractions that have a common denominator :

 3.2       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:

 x2 • 10 - (103x)     10x2 - 103x
 ————————————————  =  ———————————
        10                10     

Equation at the end of step  3  :

  (10x2 - 103x)    119
  ————————————— +  ———
       10           5 

Step  4  :

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   10x² - 103x  =   x • (10x - 103) 

Calculating the Least Common Multiple :

 5.2    Find the Least Common Multiple

      The left denominator is :       10 

      The right denominator is :       5 

      Least Common Multiple:
      10 

Calculating Multipliers :

 5.3    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 :

 5.4      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.      x • (10x-103)
   ——————————————————  =   —————————————
         L.C.M                  10      

   R. Mult. • R. Num.      119 • 2
   ——————————————————  =   ———————
         L.C.M               10   

Adding fractions that have a common denominator :

 5.5       Adding up the two equivalent fractions

 x • (10x-103) + 119 • 2     10x2 - 103x + 238
 ———————————————————————  =  —————————————————
           10                       10        

Trying to factor by splitting the middle term

 5.6     Factoring  10x² - 103x + 238 

The first term is,  10x²  its coefficient is  10 .
The middle term is,  -103x  its coefficient is  -103 .
The last term, "the constant", is  +238 

Step-1 : Multiply the coefficient of the first term by the constant   10 • 238 = 2380 

Step-2 : Find two factors of  2380  whose sum equals the coefficient of the middle term, which is   -103 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -68  and  -35 
                     10x² - 68x - 35x - 238

Step-4 : Add up the first 2 terms, pulling out like factors :
                    2x • (5x-34)
              Add up the last 2 terms, pulling out common factors :
                    7 • (5x-34)
Step-5 : Add up the four terms of step 4 :
                    (2x-7)  •  (5x-34)
             Which is the desired factorization

Final result :

  (5x - 34) • (2x - 7)
  ————————————————————
           10