Solution - Adding, subtracting and finding the least common multiple

Step  1  :

            3
 Simplify   —
            x

Equation at the end of step  1  :

        9            3     
  (x +  —) -  ((x +  —) +  2)
        8            x     

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to a whole

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

          x     x • x
     x =  —  =  —————
          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 :

 2.2       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 • x + 3     x2 + 3
 —————————  =  ——————
     x           x   

Equation at the end of step  2  :

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

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction

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

         2     2 • x
    2 =  —  =  —————
         1       x  

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(x) = x² + 3
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  3.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 3.3       Adding up the two equivalent fractions

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

Equation at the end of step  3  :

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

Step  4  :

            9
 Simplify   —
            8

Equation at the end of step  4  :

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

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  8  as the denominator :

          x     x • 8
     x =  —  =  —————
          1       8  

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions

 x • 8 + 9     8x + 9
 —————————  =  ——————
     8           8   

Equation at the end of step  5  :

  (8x + 9)    (x2 + 2x + 3)
  ———————— -  —————————————
     8              x      

Step  6  :

Trying to factor by splitting the middle term

 6.1     Factoring  x²+2x+3 

The first term is,  x²  its coefficient is  1 .
The middle term is,  +2x  its coefficient is  2 .
The last term, "the constant", is  +3 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 3 = 3 

Step-2 : Find two factors of  3  whose sum equals the coefficient of the middle term, which is   2 .

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

Calculating the Least Common Multiple :

 6.2    Find the Least Common Multiple

      The left denominator is :       8 

      The right denominator is :       x 

      Least Common Multiple:
      8x 

Calculating Multipliers :

 6.3    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 = x

   Right_M = L.C.M / R_Deno = 8

Making Equivalent Fractions :

 6.4      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.      (8x+9) • x
   ——————————————————  =   ——————————
         L.C.M                 8x    

   R. Mult. • R. Num.      (x2+2x+3) • 8
   ——————————————————  =   —————————————
         L.C.M                  8x      

Adding fractions that have a common denominator :

 6.5       Adding up the two equivalent fractions

 (8x+9) • x - ((x2+2x+3) • 8)     -7x - 24
 ————————————————————————————  =  ————————
              8x                     8x   

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   -7x - 24  =   +1 • (7x + 24) 

Final result :

 +1 • (7x + 24)