Note that, the roots of the given eqn. are,
#alpha=(-1+isqrt3)/2 and beta=(-1-isqrt3)/2#, which are the
cube roots of unity, i.e., #omega and omega^2#.
Clearly, #omega^3=1#.
Without any loss of generality, let us select,
#alpha=omega, and, beta=omega^2#.
We seek for the eqn. which has roots,
#alpha'=alpha^19, &, beta'=beta^17#.
Now, #alpha'+beta'=alpha^19+beta^17=omega^19+(omega^2)^17#,
#=omega^19+omega^34=omega^19(1+omega^15)#,
#=(omega^3)^6*omega{1+(omega^3)^5}#,
#=(1)^6*omega{1+(1)^5}............[because, omega^3=1]#.
#alpha'+beta'=2omega.........................................................(star_1)#.
#alpha'*beta'=omega^19*omega^34=omega^53#,
#=(omega^3)^17*omega^2=(1)^17*omega^2#.
#rArr alpha'*beta'=omega^2......................................................(star_2)#.
Hence, the reqd. eqn., is given by,
#x^2-(alpha'+beta')x+alpha'beta'=0, i.e., #
#x^2-2omegax+omega^2=0, or, (x-omega)^2=0#.
#:. {x-(-1+isqrt3)/2}^2=0, or, (2x+1-isqrt3)^2=0,#
is the desired eqn.
