It is given that, α,β are the roots of x2+px+q=0 α+β=−p and αβ=q since, α4,β4 are roots of x2−rx+s=0 Therefore, α4+β4=r and α4β4=s Now, x2−4qx+2q2−r=0 D=(4q)2−4(2q2−r) =16q2−8q2+4r =8q2+4r Here r=α4+β4≥0 This implies 8q2≥0 Thus, D≥0 Therefore, the equationhas two real roots.