No solutions found

Rearrange:

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

                     x^25-(-5*x)=0 

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   x²⁵ + 5x  =   x • (x²⁴ + 5) 

Trying to factor as a Sum of Cubes :

 2.2      Factoring:  x²⁴ + 5 

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 :  5  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 • (x24 + 5)  = 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  :    x²⁴+5 = 0 

 Subtract  5  from both sides of the equation : 
                      x²⁴ = -5
                     x  =  24th root of (-5) 

 The equation has no real solutions. It has 24 imaginary, or complex solutions.
 These solutions are x = 24th root of -5.00000

1.  These solutions are x = 24th root of -5.00000
2.  x = 0