Analysis of overdamped series RLC circuit (from "Foundations of Analog and Digital Circuits")
When alpha grows the expression under square root do not tend to 1-1, it tends to 1-0 or better to 

$$
\sqrt{1-\left(\frac{\omega_0}{\alpha}\right)^2}\rightarrow1-\frac{1}{2}\left(\frac{\omega_0}{\alpha}\right)^2
$$ 

it's a well known limit.

$$
a\rightarrow 0\quad\quad\Rightarrow\quad\quad\sqrt{1\pm a}\rightarrow1\pm\frac{a}{2}$$

In case 12.69 we disregard the (small) fraction against 2.

In case 12.70 the (small) fraction is the only thing remaining.