Netwon's method going from $\Delta w$ to iteration?

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I'm following this tutorial to learn about Levenberg-Maquardt and I'm going through Newton's method. I understand most of the math in that document but I can't see how the jump from 12.14 to 12.15 happens.

Here's my understanding: We have $-\textbf{g} = \textbf{H}\Delta \textbf{w}$. We got this based on the fact that at the minima, $\textbf{g}$ would be zero. Since we plugged this into the equation 12.9, we should get a value of $\Delta \textbf{w}$ that directly takes us to the minima. Why do we need to iterate multiple times?

Is there some implicit assumption that requires the iteration?

(I couldn't find a self-study tag on this forum - is it called something else here?)