Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (2x213 • x) -  24  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   2x²¹⁴ - 24  =   2 • (x²¹⁴ - 12) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  x²¹⁴ - 12 

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 : 12 is not a square !!

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

Equation at the end of step  3  :

  2 • (x214 - 12)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    2   =  0

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

Solving a Single Variable Equation :

 4.2      Solve  :    x²¹⁴-12 = 0 

 Add  12  to both sides of the equation : 
                      x²¹⁴ = 12
                     x  =  214th root of (12) 

 The equation has two real solutions  
 These solutions are  x = ± 214th root of 12 = ± 1.0117  
 

Two solutions were found :