Solution - Adding, subtracting and finding the least common multiple

Step  1  :

             9
 Simplify   ——
            x2

Equation at the end of step  1  :

        6      9
  (1 +  —) +  ——
        x     x2

Step  2  :

            6
 Simplify   —
            x

Equation at the end of step  2  :

        6      9
  (1 +  —) +  ——
        x     x2

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a fraction to a whole

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

          1     1 • x
     1 =  —  =  —————
          1       x  

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:

 x + 6     x + 6
 —————  =  —————
   x         x  

Equation at the end of step  3  :

  (x + 6)     9
  ——————— +  ——
     x       x2

Step  4  :

Calculating the Least Common Multiple :

 4.1    Find the Least Common Multiple

      The left denominator is :       x 

      The right denominator is :       x² 

      Least Common Multiple:
      x² 

Calculating Multipliers :

 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 = x

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 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.

   L. Mult. • L. Num.      (x+6) • x
   ——————————————————  =   —————————
         L.C.M                x2    

   R. Mult. • R. Num.       9
   ——————————————————  =   ——
         L.C.M             x2

Adding fractions that have a common denominator :

 4.4       Adding up the two equivalent fractions

 (x+6) • x + 9     x2 + 6x + 9
 —————————————  =  ———————————
      x2               x2     

Trying to factor by splitting the middle term

 4.5     Factoring  x² + 6x + 9 

The first term is,  x²  its coefficient is  1 .
The middle term is,  +6x  its coefficient is  6 .
The last term, "the constant", is  +9 

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

Step-2 : Find two factors of  9  whose sum equals the coefficient of the middle term, which is   6 .

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

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

Multiplying Exponential Expressions :

 4.6    Multiply  (x+3)  by  (x+3) 

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

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

Final result :

  (x + 3)2
  ————————
     x2