= Solution
For $a\ne0$, factor out $|a|$, not $a$, and apply the <binomial series> with exponent $1/2$:
$$
\sqrt{a^2+x^2}=|a|\left(1+\frac{x^2}{a^2}\right)^{1/2}=\boxed{|a|+\frac{x^2}{2|a|}-\frac{x^4}{8|a|^3}+O(x^6).}
$$
For each fixed nonzero $a$, these are the first three nonzero terms of the <Taylor series> about zero; the full <binomial series> converges when $|x|<|a|$.
For $a=0$, the <function> is $|x|$. Its right <derivative> at zero is $1$ and its left <derivative> is $-1$, so \b[no Taylor series about zero exists when $a=0$]. The one-sided formulas are exactly $x$ for $x\geq0$ and $-x$ for $x\leq0$; they do not form a single two-sided <Taylor expansion>, nor can they supply three nonzero terms. The parameter case explicitly included in the question must therefore be treated as this exception.
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