Question

if one root of x2+px+q=0 may be the square of the other,
then p3+q2​

Answer 1

Answer:

Hope it helps

Step-by-step explanation:

For quadratic equation  ax2+bx+c=0 :

sum of roots  =−b/a  

product of roots  =c/a  

If one root of the equation

x2+px+q=0  

is the square of the other, then roots are  ω  and  ω2  

sum of roots  =−p⟹p=−ω2−ω  

product of roots  =q⟹q=ω2⋅ω=ω3  

p3−q(3p−1)+q2  

=(−ω2−ω)3−ω3(3(−ω2−ω)−1)+(ω3)2  

=(−ω(ω+1))3−ω3(−3ω2−3ω−1)+ω6

=−ω3(ω3+3ω2+3ω+1)+3ω5+3ω4+ω3+ω6

=−ω6−3ω5−3ω4−ω3+3ω5+3ω4+ω3+ω6

=0