Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (22x211 • x) -  3  = 0 
 Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  4x²¹²-3 

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 :  4  is the square of  2 
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  2  :

  4x212 - 3  = 0 

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    4x²¹²-3 = 0 

 Add  3  to both sides of the equation : 
                      4x²¹² = 3
Divide both sides of the equation by 4:
                     x²¹² = 3/4 = 0.750
                     x  =  212th root of (3/4) 

 The equation has two real solutions  
 These solutions are  x = 212th root of ( 0.750) = ± 0.99864  
 

Two solutions were found :