Solution - Adding, subtracting and finding the least common multiple

Step  1  :

            1
 Simplify   —
            4

Equation at the end of step  1  :

  1   1         
  — - —) ÷ x -  4
  x   4         

Step  2  :

            1
 Simplify   —
            x

Equation at the end of step  2  :

  1   1         
  — - —) ÷ x -  4
  x   4         

Step  3  :

Calculating the Least Common Multiple :

 3.1    Find the Least Common Multiple

      The left denominator is :       x 

      The right denominator is :       4 

      Least Common Multiple:
      4x 

Calculating Multipliers :

 3.2    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 = 4

   Right_M = L.C.M / R_Deno = x

Making Equivalent Fractions :

 3.3      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.       4
   ——————————————————  =   ——
         L.C.M             4x

   R. Mult. • R. Num.       x
   ——————————————————  =   ——
         L.C.M             4x

Adding fractions that have a common denominator :

 3.4       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:

 4 - (x)     4 - x
 ———————  =  —————
   4x         4x  

Equation at the end of step  3  :

  (4 - x)        
  ——————— ÷ x -  4
    4x           

Step  4  :

         4-x      
 Divide  ———  by  x
         4x       

Multiplying exponential expressions :

 4.1    x¹ multiplied by x¹ = x^{(1 + 1)} = x²

Equation at the end of step  4  :

  (4 - x)    
  ——————— -  4
    4x2      

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  4x²  as the denominator :

         4     4 • 4x2
    4 =  —  =  ———————
         1       4x2  

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 :

 5.2       Adding up the two equivalent fractions

 (4-x) - (4 • 4x2)     -16x2 - x + 4
 —————————————————  =  —————————————
        4x2                 4x2     

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   -16x² - x + 4  =   -1 • (16x² + x - 4) 

Trying to factor by splitting the middle term

 6.2     Factoring  16x² + x - 4 

The first term is,  16x²  its coefficient is  16 .
The middle term is,  +x  its coefficient is  1 .
The last term, "the constant", is  -4 

Step-1 : Multiply the coefficient of the first term by the constant   16 • -4 = -64 

Step-2 : Find two factors of  -64  whose sum equals the coefficient of the middle term, which is   1 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  +16x2 + x + 4
  —————————————
       4x2