Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (5x213 • x) -  6  = 0 
 Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  5x²¹⁴-6 

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 :  5  is not a square !!

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

Equation at the end of step  2  :

  5x214 - 6  = 0 

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    5x²¹⁴-6 = 0 

 Add  6  to both sides of the equation : 
                      5x²¹⁴ = 6
Divide both sides of the equation by 5:
                     x²¹⁴ = 6/5 = 1.200
                     x  =  214th root of (6/5) 

 The equation has two real solutions  
 These solutions are  x = 214th root of ( 1.200) = ± 1.00085  
 

Two solutions were found :