Solution
= Solution
For $f(x)=x^2-a$, its <derivative> is $f'(x)=2x$. Substitution into <Newton's method> gives
$$
\boxed{x_n=x_{n-1}-\frac{x_{n-1}^2-a}{2x_{n-1}}
=\frac12\left(x_{n-1}+\frac a{x_{n-1}}\right).}
$$
Every positive input gives another positive input, so a starting value $x_0>0$ keeps the denominator nonzero at every step.