How can you take a square root of a negative number?

Question

How can you take a square root of a negative number?

Specifically, I have this problem where I need to select a positive integer $q$ s.t. for $b \in ]2,5[$, $$q > \sqrt{\frac{1}{b-5}}$$ and I want to be able to conclude that $$b < 5-\frac{1}{q^2}$$

And the reason that $q$ has to be selected this way is of course:

$$b < 5-\frac{1}{q^2}$$ $$bq^2 < 5q^2-1$$ $$q^2(b-5)< -1$$

and since $b-5 < 0$ ($b \in ]2,5[$)

$$q^2 > \frac{1}{b-5}$$

But how can I select such $q$?

Answer 1

You made an error on taking the reciprocals: the smaller becomes the larger.

Explicitly, \begin{align} q>\frac1{\sqrt{b-5}}&\iff q^2>\frac1{b-5}\quad\textbf{and}\quad b-5>0\\ &\iff b-5>\frac1{q^2}\quad\textbf{and}\quad b-5>0\\ &\iff 5<b<5+\frac1{q^2}. \end{align}