Binomial expansion of square root

Question

I want to find the value of the square root $$(a+ b x^2 + c x)^{0.5}$$ by knowing that $x$ is very small ($x$ goes to zero).

Could anyone help me to decide what to do? How can I do that? Is the binomial expansion a good method to find the approximate value of the square root to second order in $x$? If yes, how should I binomial expand it? $a$ could be a negative number or an imaginary number or a positive number.

Answer 1

hint

Put $bx^2+cx=at $ with $a>0$.

$$(a+bx^2+cx)^{1/2}=a^{1/2}(1+t)^{1/2} $$

$(1+t)^{1/2}=1+\frac {t}{2}+\frac {1/2 (1/2-1)}{2!}t^2+ ... +\frac {1/2 (1/2-1)... (1/2-n+1)}{n!}t^n+... $

Answer 2

Hint. Note that for $|z|<1$, $$(1+z)^{1/2}=\sum_{n=0}^{\infty}\binom{1/2}{n}z^n=1+\frac{z}{2}-\frac{z^2}{8}+o(z^2).$$ Now for $a\not=0$, the second order expansion at $0$ is $$(a+ b x^2 + c x)^{1/2}=r(1+z)^{1/2}=r\left(1+\frac{cx}{2a}+\frac{(4ab-c^2)x^2}{8a^2}+o(x^2)\right)$$ where $z=(b x^2 + c x)/a$ and $r$ is the principal square root of $a$ (or $r=\sqrt{a}$ if we are dealing with real numbers and $a>0$).