Solution - Nonlinear equations

Step by step solution :

Step  1  :

Polynomial Roots Calculator :

 1.1    Find roots (zeroes) of :       F(r) = r²+10
Polynomial Roots Calculator is a set of methods aimed at finding values of  r  for which   F(r)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  r  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  10.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,5 ,10

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  1  :

  r2 + 10  = 0 

Step  2  :

Solving a Single Variable Equation :

 2.1      Solve  :    r²+10 = 0 

 Subtract  10  from both sides of the equation : 
                      r² = -10
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      r  =  ± √ -10  

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

Accordingly,  √ -10  =
                    √ -1• 10   =
                    √ -1 •√  10   =
                    i •  √ 10

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

                      r=  0.0000 + 3.1623 i 
                      r=  0.0000 - 3.1623 i 

Two solutions were found :

1.   r=  0.0000 - 3.1623 i 
   
2.   r=  0.0000 + 3.1623 i