Solution - Nonlinear equations

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     n^2-(2000000)=0 

Step by step solution :

Step  1  :

Trying to factor as a Difference of Squares :

 1.1      Factoring:  n²-2000000 

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 :  1  is not a square !!

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

Equation at the end of step  1  :

  n2 - 2000000  = 0 

Step  2  :

Solving a Single Variable Equation :

 2.1      Solve  :    n²-2000000 = 0 

 Add  2000000  to both sides of the equation : 
                      n² = 2000000
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      n  =  ± √ 2000000  

 Can  √ 2000000 be simplified ?

Yes!   The prime factorization of  2000000   is
   2•2•2•2•2•2•2•5•5•5•5•5•5 
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).

√ 2000000   =  √ 2•2•2•2•2•2•2•5•5•5•5•5•5   =2•2•2•5•5•5•√ 2   =
                ±  1000 • √ 2

The equation has two real solutions  
 These solutions are  n = 1000 • ± √2 = ± 1414.2136  
 

Two solutions were found :