Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  2x2 +  1  = 0 

Step  2  :

Polynomial Roots Calculator :

 2.1    Find roots (zeroes) of :       F(x) = 2x²+1
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  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  2  and the Trailing Constant is  1.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  2  :

  2x2 + 1  = 0 

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    2x²+1 = 0 

 Subtract  1  from both sides of the equation : 
                      2x² = -1
Divide both sides of the equation by 2:
                     x² = -1/2 = -0.500
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ -1/2  

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

Accordingly,  √ -1/2  =
                    √ -1• 1/2   =
                    √ -1 •√  1/2   =
                    i •  √ 1/2

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

                      x=  0.0000 + 0.7071 i 
                      x=  0.0000 - 0.7071 i 

Two solutions were found :

1.   x=  0.0000 - 0.7071 i 
   
2.   x=  0.0000 + 0.7071 i