Solution - Reducing fractions to their lowest terms

Step  1  :

            a2
 Simplify   ——
            18

Equation at the end of step  1  :

         a2     
  (36 -  ——) +  3a
         18     

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  18  as the denominator :

           36     36 • 18
     36 =  ——  =  ———————
           1        18   

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:

 36 • 18 - (a2)     648 - a2
 ——————————————  =  ————————
       18              18   

Equation at the end of step  2  :

  (648 - a2)    
  —————————— +  3a
      18        

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  18  as the denominator :

          3a     3a • 18
    3a =  ——  =  ———————
          1        18   

Trying to factor as a Difference of Squares :

 3.2      Factoring:  648 - a² 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  648  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Adding fractions that have a common denominator :

 3.3       Adding up the two equivalent fractions

 (648-a2) + 3a • 18     -a2 + 54a + 648
 ——————————————————  =  ———————————————
         18                   18       

Trying to factor by splitting the middle term

 3.4     Factoring  -a² + 54a + 648 

The first term is,  -a²  its coefficient is  -1 .
The middle term is,  +54a  its coefficient is  54 .
The last term, "the constant", is  +648 

Step-1 : Multiply the coefficient of the first term by the constant   -1 • 648 = -648 

Step-2 : Find two factors of  -648  whose sum equals the coefficient of the middle term, which is   54 .

For tidiness, printing of 14 lines which failed to find two such factors, was suppressed

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

Final result :

  +a2 + 54a + 648
  ———————————————
        18