Solution - Power equations

Step  1  :

Equation at the end of step  1  :

  (((x2)-x)-12)   (((x2)+x)-6)
  ————————————— ÷ ————————————
       2x           (2•3x2)   

Step  2  :

            x2 + x - 6
 Simplify   ——————————
             (2•3x2)  

Trying to factor by splitting the middle term

 2.1     Factoring  x² + x - 6 

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

Step-1 : Multiply the coefficient of the first term by the constant   1 • -6 = -6 

Step-2 : Find two factors of  -6  whose sum equals the coefficient of the middle term, which is   1 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -2  and  3 
                     x² - 2x + 3x - 6

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

Equation at the end of step  2  :

  (((x2)-x)-12)   (x+3)•(x-2)
  ————————————— ÷ ———————————
       2x           (2•3x2)  

Step  3  :

            x2 - x - 12
 Simplify   ———————————
                2x     

Trying to factor by splitting the middle term

 3.1     Factoring  x² - x - 12 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -x  its coefficient is  -1 .
The last term, "the constant", is  -12 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -12 = -12 

Step-2 : Find two factors of  -12  whose sum equals the coefficient of the middle term, which is   -1 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -4  and  3 
                     x² - 4x + 3x - 12

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

Equation at the end of step  3  :

  (x + 3) • (x - 4)   (x + 3) • (x - 2)
  ————————————————— ÷ —————————————————
         2x                (2•3x2)     

Step  4  :

         (x+3)•(x-4)      (x+3)•(x-2)
 Divide  ———————————  by  ———————————
             2x             (2•3x2)  

 4.1    Dividing fractions

To divide fractions, write the divison as multiplication by the reciprocal of the divisor :

(x + 3) • (x - 4)     (x + 3) • (x - 2)       (x + 3) • (x - 4)          (2•3x2)     
—————————————————  ÷  —————————————————   =   —————————————————  •  —————————————————
       2x                  (2•3x2)                   2x             (x + 3) • (x - 2)

Canceling Out :

 4.2    Cancel out  (x + 3)  which appears on both sides of the fraction line.

Dividing exponential expressions :

 4.3    x² divided by x¹ = x^{(2 - 1)} = x¹ = x

Canceling Out :

 4.4      Canceling out  2  as it appears on both sides of the fraction line

Final result :

  3x • (x - 4)
  ————————————
     x - 2