Solution - Factoring binomials using the difference of squares

Rearrange:

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

                     x^28-(-6*x)=0 

Step by step solution :

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   x²⁸ + 6x  =   x • (x²⁷ + 6) 

Trying to factor as a Sum of Cubes :

 2.2      Factoring:  x²⁷ + 6 

Theory : A sum of two perfect cubes,  a³ + b³ can be factored into  :
             (a+b) • (a²-ab+b²)
Proof  : (a+b) • (a²-ab+b²) =
    a³-a²b+ab²+ba²-b²a+b³ =
    a³+(a²b-ba²)+(ab²-b²a)+b³=
    a³+0+0+b³=
    a³+b³

Check :  6  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Equation at the end of step  2  :

  x • (x27 + 6)  = 0 

Step  3  :

Theory - Roots of a product :

 3.1    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 :

 3.2      Solve  :    x = 0 

  Solution is  x = 0

Solving a Single Variable Equation :

 3.3      Solve  :    x²⁷+6 = 0 

 Subtract  6  from both sides of the equation : 
                      x²⁷ = -6
                     x  =  27th root of (-6) 

 Negative numbers have real 27th roots.
 27th root of (-6) = ²⁷√ -1• 6  = ²⁷√ -1 • ²⁷√ 6  =(-1)•²⁷√ 6 

The equation has one real solution, a negative number This solution is  x = negative 27th root of 6 = -1.0686

Two solutions were found :

1.  x = negative 27th root of 6 = -1.0686
2.  x = 0