Before you tackle this problem see
In this problem you must first observe the path of the roots of the
quadratic equations $x^2 + px + q = 0$ as you change $p$ and keep
You can change the equation $x^2 + px +q = 0$ by moving the point
$(p, q)$ in the red frame. You can see how the graph of $y=x^2 + px
+ q$ changes in the blue frame. The Argand Diagram in the green
frame shows the roots of the quadratic equation. Look for two roots
in the Argand diagram and watch them move as you change the driving
point $(p,q)$ in the red frame, and in doing so change the
quadratic equation and its roots.
What do you notice about the paths that these roots follow when you
change $p$ and keep $q$ fixed? Make a conjecture about the curves
on which the complex roots lie.
Now prove your conjecture.
If you have only just met complex numbers then follow the
steps given in
to see if your conjecture was correct.