Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  22x2 +  9  = 0 

Step  2  :

Polynomial Roots Calculator :

 2.1    Find roots (zeroes) of :       F(x) = 4x²+9
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  4  and the Trailing Constant is  9.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1 ,3 ,9

 Let us test ....

Note - For tidiness, printing of 13 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Equation at the end of step  2  :

  4x2 + 9  = 0 

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    4x²+9 = 0 

 Subtract  9  from both sides of the equation : 
                      4x² = -9
Divide both sides of the equation by 4:
                     x² = -9/4 = -2.250
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ -9/4  

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

Accordingly,  √ -9/4  =
                    √ -1• 9/4   =
                    √ -1 •√  9/4   =
                    i •  √ 9/4

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

                      x=  0.0000 + 1.5000 i 
                      x=  0.0000 - 1.5000 i 

Two solutions were found :

1.   x=  0.0000 - 1.5000 i 
   
2.   x=  0.0000 + 1.5000 i