Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D;:;A{ ightarrow}A$ such that $D(x)[D(x),x]^2;{in};rad(A)$ or $[D(x), x]^2 D(x);{in};rad(A)$ for all $x;{in} A$. In this case, we have $D(A);{subseteq};rad(A)$.