Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((2•3x27) • x) -  3  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   6x²⁸ - 3  =   3 • (2x²⁸ - 1) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  2x²⁸ - 1 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  2  is not a square !!

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

Equation at the end of step  3  :

  3 • (2x28 - 1)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    3   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 4.2      Solve  :    2x²⁸-1 = 0 

 Add  1  to both sides of the equation : 
                      2x²⁸ = 1
Divide both sides of the equation by 2:
                     x²⁸ = 1/2 = 0.500
                     x  =  28th root of (1/2) 

 The equation has two real solutions  
 These solutions are  x = 28th root of ( 0.500) = ± 0.97555  
 

Two solutions were found :