Quadratic Equation and Its Root

 

Definition:    The equation ax² + bx + c = 0 is known as quadratic equation for x when a  0.

 

 

Solution to ax² + bx + c = 0 :

                                                a( x² +  x ) + c = 0

                                   a( x² + 2  x ) + c = 0

                                   a(x² + 2  x  + (  )² ) + c = a (  )²

                                           a ( x +  )²   = a (  )² -  c

                                 ( x +  )²   =  

 

                    Case I:    b² – 4ac  0

                                   Then

                                               x =

 

                                    if   b² – 4ac = 0,  it can be simplified as

                                                    x =   ,   (   repeated roots )

 

                    Case II:   b² – 4ac < 0

                                    In this case,    < 0 .

                                    x has no real root.

 

                              However, if the concept of “imaginary numbers” or “complex numbers” are introduced,  the equation has roots on complex plane.  In other words,  if  i =   is defined, then

                                                    x =

 

 

                             This equation can be seen in many places when solving other problems, for example, the solution to the second order differential equations,  or difference equations.

 

                             The concept of imaginary numbers or complex numbers is also started from finding the roots of the equation.

 

 

Definition: ( Real Number )

                        The definition of real number can be considered in the following way:

                     A number r is a real number if  there exists a such that 

 

                                                          a  0,  r² = a

 

 

                      Usually, we use R to denote the set consisting of all real numbers.

                      Then 1, 2, 3, -3,  are real numbers.

 

Definition: ( Imaginary number )

                    A number b is an imaginary number if there exists a such that

                                                   a < 0,  b² = a .

 

 

                    It always assumes the form   ri  where  i =   and r is a real number.

                   

 

Definition: (  Complex number )

                    Complex number has the form

                                                       a + b i

                                   where a and b are real numbers.

 

                     Usually, we use C to denote the set consisting of all complex numbers.  Complex numbers can be treated as “an extension field of real numbers”  such that all the quadratic equations with real coefficients will have roots in this “extension field”.