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  • If alpha , beta , gamma be the roots of x^3+27=0 then the quadratic equation having roots alpha^-2 beta ^-2 and alpha^-2 gamma^2 we have

If alpha , beta , gamma be the roots of x^3+27=0 then the quadratic equation having roots alpha^-2 beta ^-2 and alpha^-2 gamma^2 we have

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Solution:    We have ,  x^3+27=0

                  alpha+eta+gamma=0  .............(1)

                  (alpha eta)+(eta gamma)+(gammaalpha)=0..........(2) ,    (alphaeta gamma)=-27 ....(3)

     Now  , sum of the roots   

     =alpha^-2eta^2+alpha^-2gamma^2=fraceta^2+gamma^2alpha^2=frac(eta+gamma)^2-2eta gamma9=frac9-189=-1

Product of the root  alpha^-2eta^2alpha^-2gamma^-2=frac(eta gamma)^2alpha^4=1

herefore      Equation is x^2+x+1=0 , whose roots are omega and   omega^2

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Deependra Verma

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