Solution - Nonlinear equations

Rearrange:

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

                     5*n^2-7-(488)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (5n2 -  7) -  488  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   5n² - 495  =   5 • (n² - 99) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  n² - 99 

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 : 99 is not a square !!

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

Equation at the end of step  3  :

  5 • (n2 - 99)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    5   =  0

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

Solving a Single Variable Equation :

 4.2      Solve  :    n²-99 = 0 

 Add  99  to both sides of the equation : 
                      n² = 99
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      n  =  ± √ 99  

 Can  √ 99 be simplified ?

Yes!   The prime factorization of  99   is
   3•3•11 
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).

√ 99   =  √ 3•3•11   =
                ±  3 • √ 11

The equation has two real solutions  
 These solutions are  n = 3 • ± √11 = ± 9.9499  
 

Two solutions were found :