Using Taylor Polynomial to Show How An Expression Of Only Real Numbers Can Be Approximated

Question

I am studying for my graduate level GQE and looking at problems from old exams. The following question (from an unknown original source) reads:

Suppose a,b,c and d are positive real numbers with a $>$ d. Use an appropriate Taylor polynomial to show that if

$\Big ( c - \frac{cd}{a} \Big )^{2} \gg \frac{4bc}{a}$

then the expression

$z = \sqrt{\Big ( c - \frac{cd}{a} \Big )^{2} + \frac{4bc}{a}}$

can be approximated by

$z \approx c - \frac{cd}{a} + \frac{2b}{a-d}$.

All Taylor Approximation problems that I have been introduced to involve a variable and an "a = ..." value. My thoughts are that $z \approx c - \frac{cd}{a} + \frac{2b}{a-d}$ resembles a second degree Taylor polynomial. How do I choose an appropriate Taylor polynomial without the typical givens? I would just like direction towards a theorem I have yet to learn or a hint in what my thought process should be regarding the structure of the question. Thank you!

Answer 1

Hint: Let $f(x)=\sqrt{p^2 +x}$, where $p\gt 0$. The first degree Maclaurin polynomial for $f(x)$ is $p+\frac{1}{2p}x$.