Solution - Nonlinear equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x3"   was replaced by   "x^3". 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (3 • (x2)) -  (22•32x33)  = 0 
 Step  2  :

Equation at the end of step  2  :

  3x2 -  (22•32x33)  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   3x² - 36x³³  =   -3x² • (12x³¹ - 1) 

Equation at the end of step  4  :

  -3x2 • (12x31 - 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  :    -3x² = 0 

 Multiply both sides of the equation by (-1) :  3x² = 0


Divide both sides of the equation by 3:
                     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  :    12x³¹-1 = 0 

 Add  1  to both sides of the equation : 
                      12x³¹ = 1
Divide both sides of the equation by 12:
                     x³¹ = 1/12 = 0.083
                     x  =  31st root of (1/12) 

 The equation has one real solution
This solution is  x = 31st root of ( 0.083) = 0.92297

Two solutions were found :

1.  x = 31st root of ( 0.083) = 0.92297
2.  x = 0