Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  32x2 -  384  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   9x² - 384  =   3 • (3x² - 128) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  3x² - 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 :  3  is not a square !!

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

Equation at the end of step  3  :

  3 • (3x2 - 128)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    3   =  0

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

Solving a Single Variable Equation :

 4.2      Solve  :    3x²-128 = 0 

 Add  128  to both sides of the equation : 
                      3x² = 128
Divide both sides of the equation by 3:
                     x² = 128/3 = 42.667
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 128/3  

 The equation has two real solutions  
 These solutions are  x = ±√ 42.667 = ± 6.53197  
 

Two solutions were found :