Solution - Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (0 -  5x2) -  3  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   -5x² - 3  =   -1 • (5x² + 3) 

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(x) = 5x² + 3
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  5  and the Trailing Constant is  3.

 The factor(s) are:

of the Leading Coefficient :  1,5
 of the Trailing Constant :  1 ,3

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  -5x2 - 3  = 0 

Step  4  :

Solving a Single Variable Equation :

 4.1      Solve  :    -5x²-3 = 0 

 Add  3  to both sides of the equation : 
                      -5x² = 3
Multiply both sides of the equation by (-1) :  5x² = -3


Divide both sides of the equation by 5:
                     x² = -3/5 = -0.600
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ -3/5  

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

Accordingly,  √ -3/5  =
                    √ -1• 3/5   =
                    √ -1 •√  3/5   =
                    i •  √ 3/5

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

                      x=  0.0000 + 0.7746 i 
                      x=  0.0000 - 0.7746 i 

Two solutions were found :

1.   x=  0.0000 - 0.7746 i 
   
2.   x=  0.0000 + 0.7746 i