Solution - Nonlinear equations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "r2"   was replaced by   "r^2". 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (r2) -  (22•3r23)  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   r² - 12r²³  =   -r² • (12r²¹ - 1) 

Trying to factor as a Difference of Cubes:

 3.2      Factoring:  12r²¹ - 1 

Theory : A difference 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 :  12  is not a cube !!

Ruling : Binomial can not be factored as the difference of two perfect cubes

Equation at the end of step  3  :

  -r2 • (12r21 - 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  :    -r² = 0 

 Multiply both sides of the equation by (-1) :  r² = 0


 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      r  =  ± √ 0  

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

Solving a Single Variable Equation :

 4.3      Solve  :    12r²¹-1 = 0 

 Add  1  to both sides of the equation : 
                      12r²¹ = 1
Divide both sides of the equation by 12:
                     r²¹ = 1/12 = 0.083
                     r  =  21st root of (1/12) 

 The equation has one real solution
This solution is  r = 21st root of ( 0.083) = 0.88840

Two solutions were found :

1.  r = 21st root of ( 0.083) = 0.88840
2.  r = 0