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^2-2-(-92)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (5n2 -  2) -  -92  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   5n² + 90  =   5 • (n² + 18) 

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(n) = n² + 18
Polynomial Roots Calculator is a set of methods aimed at finding values of  n  for which   F(n)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  n  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  18.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,6 ,9 ,18

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  5 • (n2 + 18)  = 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+18 = 0 

 Subtract  18  from both sides of the equation : 
                      n² = -18
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      n  =  ± √ -18  

 In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 

Accordingly,  √ -18  =
                    √ -1• 18   =
                    √ -1 •√  18   =
                    i •  √ 18

Can  √ 18 be simplified ?

Yes!   The prime factorization of  18   is
   2•3•3 
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).

√ 18   =  √ 2•3•3   =
                ±  3 • √ 2

The equation has no real solutions. It has 2 imaginary, or complex solutions.

                      n=  0.0000 + 4.2426 i 
                      n=  0.0000 - 4.2426 i 

Two solutions were found :

1.   n=  0.0000 - 4.2426 i 
   
2.   n=  0.0000 + 4.2426 i