Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  23x2 -  4  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   8x² - 4  =   4 • (2x² - 1) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  2x² - 1 

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 :  2  is not a square !!

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

Equation at the end of step  3  :

  4 • (2x2 - 1)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    4   =  0

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

Solving a Single Variable Equation :

 4.2      Solve  :    2x²-1 = 0 

 Add  1  to both sides of the equation : 
                      2x² = 1
Divide both sides of the equation by 2:
                     x² = 1/2 = 0.500
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 1/2  

 The equation has two real solutions  
 These solutions are  x = ±√ 0.500 = ± 0.70711  
 

Two solutions were found :