Solution - Nonlinear equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x6"   was replaced by   "x^6". 

Rearrange:

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

                     5*x^214*x-(x^6)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (5x214 • x) -  x6  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   5x²¹⁵ - x⁶  =   x⁶ • (5x²⁰⁹ - 1) 

Equation at the end of step  3  :

  x6 • (5x209 - 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  :    x⁶ = 0 

  Solution is  x⁶ = 0

Solving a Single Variable Equation :

 4.3      Solve  :    5x²⁰⁹-1 = 0 

 Add  1  to both sides of the equation : 
                      5x²⁰⁹ = 1
Divide both sides of the equation by 5:
                     x²⁰⁹ = 1/5 = 0.200
                     x  =  209th root of (1/5) 

 The equation has one real solution
This solution is  x = 209th root of ( 0.200) = 0.99233

Two solutions were found :

1.  x = 209th root of ( 0.200) = 0.99233
2.  x⁶ = 0