If α and β are the roots of the equation x2+px+q=0 ,  and α4 and β4 are the roots of x2−rx+q=0, the roots of x2−4qx+2q2−r=0 are always

α,β are roots of x2+px+q=0⇒ α+β=−p and αβ=qα4,β4 are roots of x2−rx+q=0⇒ α4+β4=r and α4β4=q Now for equation x2−4qx+2q2−r=0 , Product of roots=2q2−r=2(αβ)2−α4+β4=−α2−β22<0Product of roots is negative then roots must be real.