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\begin{aligned} x_{n+1} &=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}=x_{n}-\frac{x_{n}^{2}-5}{2 x_{n}}=\frac{2 x_{n}^{2}-\left(x_{n}^{2}-5\right)}{2 x_{n}} \\ &=\frac{x_{n}^{2}+5}{2 x_{n}}=\frac{1}{2}\left\{x_{n}+\frac{5}{x_{n}}\right\} \end{aligned}