If #alpha and beta# are the roots of a quadratic equation #px^2+qx+q=0#, prove that #sqrt(alpha/beta)+sqrt(beta/alpha)+sqrt(q/p)=0#?

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If #alpha and beta# are the roots of a quadratic equation #px^2+qx+q=0#, prove that #sqrt(alpha/beta)+sqrt(beta/alpha)+sqrt(q/p)=0#

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Answer 1 · 2018-04-09T07:20:32

Please see below.

As #alpha# and#beta# are roots of #px^2+qx+q=0#,

we have #alpha+beta=-q/p# and #alphabeta=q/p#

Hence #sqrt(alpha/beta)+sqrt(beta/alpha)+sqrt(q/p)#

= #sqrtalpha/sqrtbeta+sqrtbeta/sqrtalpha+sqrt(q/p)#

= #(alpha+beta)/sqrt(alphabeta)+sqrt(q/p)#

= #(-q/p)/sqrt(q/p)+sqrt(q/p)#

= #-sqrt(q/p)+sqrt(q/p)#

= #0#