This is a Test of Mathematics Solution Subjective 73 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

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## 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.