Solution - Nonlinear equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2". 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (4 • (x2)) -  (22•5x25)  = 0 
 Step  2  :

Equation at the end of step  2  :

  22x2 -  (22•5x25)  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   4x² - 20x²⁵  =   -4x² • (5x²³ - 1) 

Equation at the end of step  4  :

  -4x2 • (5x23 - 1)  = 0 

Step  5  :

Theory - Roots of a product :

 5.1    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 :

 5.2      Solve  :    -4x² = 0 

 Multiply both sides of the equation by (-1) :  4x² = 0


Divide both sides of the equation by 4:
                     x² = 0
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 0  

 Any root of zero is zero. This equation has one solution which is  x = 0

Solving a Single Variable Equation :

 5.3      Solve  :    5x²³-1 = 0 

 Add  1  to both sides of the equation : 
                      5x²³ = 1
Divide both sides of the equation by 5:
                     x²³ = 1/5 = 0.200
                     x  =  23rd root of (1/5) 

 The equation has one real solution
This solution is  x = 23rd root of ( 0.200) = 0.93242

Two solutions were found :

1.  x = 23rd root of ( 0.200) = 0.93242
2.  x = 0