Solution - Other Factorizations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "p1"   was replaced by   "p^1". 

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     4*p^26*p^1-(2*p)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (22p26 • p) -  2p  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   4p²⁷ - 2p  =   2p • (2p²⁶ - 1) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  2p²⁶ - 1 

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 :  2  is not a square !!

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

Equation at the end of step  3  :

  2p • (2p26 - 1)  = 0 

Step  4  :

Theory - Roots of a product :

 4.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 4.2      Solve  :    2p = 0 

 Divide both sides of the equation by 2:
                     p = 0

Solving a Single Variable Equation :

 4.3      Solve  :    2p²⁶-1 = 0 

 Add  1  to both sides of the equation : 
                      2p²⁶ = 1
Divide both sides of the equation by 2:
                     p²⁶ = 1/2 = 0.500
                     p  =  26th root of (1/2) 

 The equation has two real solutions  
 These solutions are  p = 26th root of ( 0.500) = ± 0.97369  
 

Three solutions were found :

1.  p = 26th root of ( 0.500) = ± 0.97369
2.  p = 0