Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (3x25 • x) -  2  = 0 
 Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  3x²⁶-2 

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 :  3  is not a square !!

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

Equation at the end of step  2  :

  3x26 - 2  = 0 

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    3x²⁶-2 = 0 

 Add  2  to both sides of the equation : 
                      3x²⁶ = 2
Divide both sides of the equation by 3:
                     x²⁶ = 2/3 = 0.667
                     x  =  26th root of (2/3) 

 The equation has two real solutions  
 These solutions are  x = 26th root of ( 0.667) = ± 0.98453  
 

Two solutions were found :