Locate the Complex roots by a quadratic graph

 

        We can find the roots of a quadratic equation:

               

by plotting a quadratic graph:

               

        The graph cuts the x-axis and the point(s) of intersection of the graph and the x-axis are the roots of the quadratic equation.

 

        However, this method is only good for real roots.

 

        Can we locate the complex roots after plotting the graph?

We can see easily that the quadratic graph cannot cut the x-axis. What can we do?

 

 

An example

 

Solve the equation :

           x2 – 2x + 5 = 0

The graph of y = x2 – 2x + 5 is given in red.

It cannot cut the x-axis.

Follow the steps to find the complex roots:

(1)     Draw the line of symmetry (in blue )

(2)     This line cuts the x-axis at A, in this case (1,0).

(3)     This line also cuts the quadratic graph at B, the

           minimum point, in this case (1,4).

(4)    Construct the point C on the line of symmetry

          such that AB = BC.

(5)     Draw the horizontal line which passes through C

          (in black)

(6)     Use  A  as center, CD as radius, draw a circle.

(7)     The circle cuts the line of symmetry at the points

           X, Y, in this case (1,2) and (1,-2).

(8)     The complex roots are 1+2i and 1-2i, which

           are given by the points X, Y in the Argand diagram.

 

 

 

 

Exercise if you want the complete story

           Check that for  D = b2 – 4ac < 0

                   (1)  

                   (2)  

                   (3)