Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  2x2 -  216  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   2x² - 216  =   2 • (x² - 108) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  x² - 108 

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 : 108 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 • (x2 - 108)  = 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  :    x²-108 = 0 

 Add  108  to both sides of the equation : 
                      x² = 108
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 108  

 Can  √ 108 be simplified ?

Yes!   The prime factorization of  108   is
   2•2•3•3•3 
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).

√ 108   =  √ 2•2•3•3•3   =2•3•√ 3   =
                ±  6 • √ 3

The equation has two real solutions  
 These solutions are  x = 6 • ± √3 = ± 10.3923  
 

Two solutions were found :