What is a good approximation of $(1-p)(1-q)$ as $(1-x)^2$, for $p,q \in (0,1)$?

Question

I'm doing some scientific modeling, and I want to use $(1-x)^2$ to approximate $(1-p)(1-q)$, with $p, q \in (0,1)$. $p$ and $q$ are probabilities, and are not near zero. My intuition is that since I'm approximating a product, the geometric mean $x=\sqrt{pq}$ is a good choice--better, at least, than the average $\frac{pq}{2}$. Does this seem sensible? Are there other recommendations you'd suggest considering? (Is the question too vague?)

Answer 1

There is an exact solution: $x = 1 - \sqrt{(1-p)(1-q)}$.