Solution - Reducing fractions to their lowest terms

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "1.5" was replaced by "(15/10)".

Step  1  :

            3
 Simplify   —
            2

Equation at the end of step  1  :

    3                  
  ((— • t2) +  5t) -  84
    2                  

Step  2  :

Equation at the end of step  2  :

   3t2           
  (——— +  5t) -  84
    2            

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction

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

          5t     5t • 2
    5t =  ——  =  ——————
          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:

 3t2 + 5t • 2     3t2 + 10t
 ————————————  =  —————————
      2               2    

Equation at the end of step  3  :

  (3t2 + 10t)    
  ——————————— -  84
       2         

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

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

          84     84 • 2
    84 =  ——  =  ——————
          1        2   

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   3t² + 10t  =   t • (3t + 10) 

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions

 t • (3t+10) - (84 • 2)     3t2 + 10t - 168
 ——————————————————————  =  ———————————————
           2                       2       

Trying to factor by splitting the middle term

 5.3     Factoring  3t² + 10t - 168 

The first term is,  3t²  its coefficient is  3 .
The middle term is,  +10t  its coefficient is  10 .
The last term, "the constant", is  -168 

Step-1 : Multiply the coefficient of the first term by the constant   3 • -168 = -504 

Step-2 : Find two factors of  -504  whose sum equals the coefficient of the middle term, which is   10 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -18  and  28 
                     3t² - 18t + 28t - 168

Step-4 : Add up the first 2 terms, pulling out like factors :
                    3t • (t-6)
              Add up the last 2 terms, pulling out common factors :
                    28 • (t-6)
Step-5 : Add up the four terms of step 4 :
                    (3t+28)  •  (t+6)
             Which is the desired factorization

Final result :

(3t+28)  •  (t+6)