Solution - Nonlinear equations

Rearrange:

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

                     2*x^29*12-(x^2)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (2x29 • 12) -  x2  = 0 

Step  2  :

Multiplying exponents :

 2.1    2¹  multiplied by  2²   = 2^{(1 + 2)} = 2³

Equation at the end of step  2  :

  (23•3x29) -  x2  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   24x²⁹ - x²  =   x² • (24x²⁷ - 1) 

Trying to factor as a Difference of Cubes:

 4.2      Factoring:  24x²⁷ - 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 :  24  is not a cube !!

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

Equation at the end of step  4  :

  x2 • (24x27 - 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  :    x² = 0 

  Solution is  x² = 0

Solving a Single Variable Equation :

 5.3      Solve  :    24x²⁷-1 = 0 

 Add  1  to both sides of the equation : 
                      24x²⁷ = 1
Divide both sides of the equation by 24:
                     x²⁷ = 1/24 = 0.042
                     x  =  27th root of (1/24) 

 The equation has one real solution
This solution is  x = 27th root of ( 0.042) = 0.88896

Two solutions were found :

1.  x = 27th root of ( 0.042) = 0.88896
2.  x² = 0