Test of Mathematics Solution Subjective 73 - Coefficients of a Quadratic

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.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

Problem

Consider the equation ${x^3 + Gx + H = 0}$ , 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 ${x^3 + Gx + H = 0}$ are

${\alpha, \beta, \gamma}$ [ Let ${\alpha, \beta}$ are complex conjugates] .Now ${\alpha \beta \gamma}$ = - H ... (i)

${\alpha + \beta +\gamma}$ = 0 ... (ii)

${\alpha \beta + \beta \gamma + \gamma \alpha}$ = G ... (iii)

From (ii) we get ${\alpha + \beta + \gamma}$

= 0 [ ${\alpha, \beta}$ are complex conjugates so they are real]

${\Rightarrow}$ ${\gamma}$ = real Now as ${\gamma}$ = real

${\beta \gamma + \gamma \alpha}$ = ${\gamma (\beta + \alpha)}$ = ${real \times real}$ = ${real}$ ... (iv) ${\alpha, \beta}$ are complex conjugates so ${\alpha \beta = real}$ ... (v) From (iv) & (v) we get

${\alpha \beta + \beta \gamma + \gamma \alpha}$ = ${real + real = real}$ ${\Rightarrow}$ G = real [from (iii)] Now ${\alpha, \beta}$ is real and ${\alpha, \beta}$ is real so ${\alpha \beta \gamma}$ = real ${\Rightarrow}$ H = real.

Related

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.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

Problem

Consider the equation ${x^3 + Gx + H = 0}$ , 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 ${x^3 + Gx + H = 0}$ are

${\alpha, \beta, \gamma}$ [ Let ${\alpha, \beta}$ are complex conjugates] .Now ${\alpha \beta \gamma}$ = - H ... (i)

${\alpha + \beta +\gamma}$ = 0 ... (ii)

${\alpha \beta + \beta \gamma + \gamma \alpha}$ = G ... (iii)

From (ii) we get ${\alpha + \beta + \gamma}$

= 0 [ ${\alpha, \beta}$ are complex conjugates so they are real]

${\Rightarrow}$ ${\gamma}$ = real Now as ${\gamma}$ = real

${\beta \gamma + \gamma \alpha}$ = ${\gamma (\beta + \alpha)}$ = ${real \times real}$ = ${real}$ ... (iv) ${\alpha, \beta}$ are complex conjugates so ${\alpha \beta = real}$ ... (v) From (iv) & (v) we get

${\alpha \beta + \beta \gamma + \gamma \alpha}$ = ${real + real = real}$ ${\Rightarrow}$ G = real [from (iii)] Now ${\alpha, \beta}$ is real and ${\alpha, \beta}$ is real so ${\alpha \beta \gamma}$ = real ${\Rightarrow}$ H = real.

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