Solution - Nonlinear equations

Step by step solution :

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  x²-128 

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 : 128 is not a square !!

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

Equation at the end of step  1  :

  x2 - 128  = 0 

Step  2  :

Solving a Single Variable Equation :

 2.1      Solve  :    x²-128 = 0 

 Add  128  to both sides of the equation : 
                      x² = 128
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 128  

 Can  √ 128 be simplified ?

Yes!   The prime factorization of  128   is
   2•2•2•2•2•2•2 
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 128   =  √ 2•2•2•2•2•2•2   =2•2•2•√ 2   =
                ±  8 • √ 2

The equation has two real solutions  
 These solutions are  x = 8 • ± √2 = ± 11.3137  
 

Two solutions were found :