problem: Consider the equation {\displaystyle{x^3 + Gx + H = 0}} , 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 {\displaystyle{x^3 + Gx + H = 0}}

are {\displaystyle{\alpha, \beta, \gamma}}                         [ Let {\displaystyle{\alpha, \beta}} are complex conjugates]

Now {\displaystyle{\alpha \beta \gamma}} = – H … (i)
{\displaystyle{\alpha + \beta + \gamma}} = 0 … (ii)
{\displaystyle{\alpha \beta + \beta \gamma + \gamma \alpha}} = G … (iii)

From (ii) we get
{\displaystyle{\alpha + \beta + \gamma}} = 0 [ {\displaystyle{\alpha, \beta}} are complex conjugates so they are real]
{\Rightarrow} {\gamma} = real

Now as {\gamma} = real

{\displaystyle{\beta \gamma + \gamma \alpha}} = {\displaystyle{\gamma (\beta + \alpha)}}
= {real \times real}
= {real} … (iv)
{\alpha, \beta} are complex conjugates so {\alpha \beta = real} … (v)
From (iv) & (v) we get {\displaystyle{\alpha \beta + \beta \gamma + \gamma \alpha}} = {real + real = real}
{\Rightarrow} G = real [from (iii)]

Now {\alpha, \beta} is real and {\alpha, \beta} is real
so {\displaystyle{\alpha \beta \gamma}} = real
{\Rightarrow} H = real.