Statement of a general form of Hensel's lemma

Question

Let $R$ be a complete ring with respect to ideal $I$ and $p \in R[x]$ polynomial. Element $a \in R$ is an approximate root of $p$ if $$ p(a) \in p'(a)^2 I. $$ One of the general forms of Hensel's lemma states that if there is an approximate root then there is a root $b$ of $p$ such that $a-b \in p'(a)I$. I understand the proof of this fact (see Eisenbud's textbook on commutative algebra), but I'm confused by the condition $p(a) \in p'(a)^2 I$. Why this is a natural condition to put in this statement?