Newton Raphson Method Iteration Scheme

Question

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My question here is for the 2nd part. The 1st part is straightforward, taking $x^2 - N = 0$ as $f(x)$. How does one go about the second part? What exactly do they mean by applying the scheme two times?

Edit: By second part, I mean how does one derive that formula "easily"!

Answer 1

If $A_k=x_k$ and $B_k=N/x_k$, then $$ x_2=\frac{A_1+B_1}2=\frac{x_1}2+\frac{N}{2x_1}=\frac{A_0+B_0}4+\frac{N}{A_0+B_0} $$ And as $x_2$ is indeed a better approximation for $\sqrt N$, one can write $\sqrt{N}\approx x_2=...$