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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

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August 12, 2015

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