= Solution
The <Newton root-finding iteration> for this <quadratic polynomial> is
$$
x_{k+1}=x_k-\frac{x_k^2-2\varepsilon x_k-1}{2(x_k-\varepsilon)}
=\frac{x_k^2+1}{2(x_k-\varepsilon)}.
$$
At $x_0=1$, it gives $x_1=1/(1-\varepsilon)$. Its <geometric series> expansion is $1+\varepsilon+\varepsilon^2+\varepsilon^3+O(\varepsilon^4)$. Subtracting the expansion of the <polynomial root> yields
$$
x_1-x_*=\frac12\varepsilon^2+\varepsilon^3+O(\varepsilon^4),
\qquad\boxed{|x_1-x_*|\sim\frac12|\varepsilon|^2,\qquad n_1=2.}
$$
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