A quadratic equation whose roots are (γα)2  and (βα)2 , where α,β,γ  are roots of x3+27=0  is

Given that, x3+27=0 ⇒x3=(−3)3 ⇒x=−3,−3ω,−3ω2 ∴  γα=−3ω2−3⇒γα=ω2, And βα=−3ω−3⇒βα=ω ∴  (γα)2=ω4=ω  and (βα)2=ω2 sum of the roots : (γα)2 +(βα)2  =ω+ω2 product of the roots : (γα)2  (βα)2 =ω3 ∴  The required equation is x2−((γα)2+(βα)2)x+(γα)2(βα)2=0 ⇒x2−(ω+ω2)x+ω3=0 ⇒x2+x+1=0 .