# Coefficients of a quadratic (Tomato subjective 73)

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.