Solution - Adding, subtracting and finding the least common multiple

Step  1  :

             9
 Simplify   ——
            16

Equation at the end of step  1  :

            3           9
  ((x2) -  (— • x)) +  ——
            2          16

Step  2  :

            3
 Simplify   —
            2

Equation at the end of step  2  :

            3           9
  ((x2) -  (— • x)) +  ——
            2          16
 Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  2  as the denominator :

           x2     x2 • 2
     x2 =  ——  =  ——————
           1        2   

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 • 2 - (3x)     2x2 - 3x
 —————————————  =  ————————
       2              2    

Equation at the end of step  3  :

  (2x2 - 3x)     9
  —————————— +  ——
      2         16

Step  4  :

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   2x² - 3x  =   x • (2x - 3) 

Calculating the Least Common Multiple :

 5.2    Find the Least Common Multiple

      The left denominator is :       2 

      The right denominator is :       16 

      Least Common Multiple:
      16 

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 = 8

   Right_M = L.C.M / R_Deno = 1

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 • (2x-3) • 8
   ——————————————————  =   ——————————————
         L.C.M                   16      

   R. Mult. • R. Num.       9
   ——————————————————  =   ——
         L.C.M             16

Adding fractions that have a common denominator :

 5.5       Adding up the two equivalent fractions

 x • (2x-3) • 8 + 9     16x2 - 24x + 9
 ——————————————————  =  ——————————————
         16                   16      

Trying to factor by splitting the middle term

 5.6     Factoring  16x² - 24x + 9 

The first term is,  16x²  its coefficient is  16 .
The middle term is,  -24x  its coefficient is  -24 .
The last term, "the constant", is  +9 

Step-1 : Multiply the coefficient of the first term by the constant   16 • 9 = 144 

Step-2 : Find two factors of  144  whose sum equals the coefficient of the middle term, which is   -24 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -12  and  -12 
                     16x² - 12x - 12x - 9

Step-4 : Add up the first 2 terms, pulling out like factors :
                    4x • (4x-3)
              Add up the last 2 terms, pulling out common factors :
                    3 • (4x-3)
Step-5 : Add up the four terms of step 4 :
                    (4x-3)  •  (4x-3)
             Which is the desired factorization

Multiplying Exponential Expressions :

 5.7    Multiply  (4x-3)  by  (4x-3) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (4x-3)  and the exponents are :
          1 , as  (4x-3)  is the same number as  (4x-3)¹ 
 and   1 , as  (4x-3)  is the same number as  (4x-3)¹ 
The product is therefore,  (4x-3)⁽¹⁺¹⁾ = (4x-3)² 

Final result :

  (4x - 3)2
  —————————
     16