* problem*: Consider the equation , where G and H are complex numbers. Suppose that this equation has a pair of complex conjugate roots. Show that both G and H are real.

* solution*: Let three roots of the equation

are [ Let are complex conjugates]

Now = – H … (i)

= 0 … (ii)

= G … (iii)

From (ii) we get

= 0 [ are complex conjugates so they are real]

= real

Now as = real

=

=

= … (iv)

are complex conjugates so … (v)

From (iv) & (v) we get =

G = real [from (iii)]

Now is real and is real

so = real

H = real.