# Simple case Newton Method quadratic convergence geometric interpretation

https://youtu.be/4Q37iOyBq44 (5:32 time)

In this video professor refers to why is $$E_1 \approx E^2_0$$ implying there is somewhat geometric interpretation in this case. It feels natural to me, but when I tried to write this down, I understood that I can't produce logic steps to this conclusion.

My guess is: because we use tangent line as approximation of our quadratic function, it only natural that (in this case), we have to land close to our desired $$x$$ and if we take $$|x-x_0|$$ as our $$\Delta x$$ on this interval it will grow as square compare to growing from $$x-\Delta x$$ to $$x$$. So if our tangent line approximation is good enough we will land close to $$x$$, therefore our $$y_1 = x_1^2; y_1 \approx \sqrt(y_0) , y_0 = x_0^2$$

Could someone please explain this to me?