In the textbook "Macroeconomic Theory" of Wickens, p. 372, I found the following approximation:
$$\frac{1}{2} \text{tr} A \pm \frac{1}{2} [(\text{tr}A)^2 - 4\text{det}A ]^{\frac{1}{2}}\approx \left \lbrace \frac {\text{det}A}{\text{tr}A}, \text{tr}A - \frac {\text{det}A}{\text{tr}A} \right \rbrace $$
which is supposedly derived from a first order Taylor series approximation. Any idea of how this result can be obtained? Any suggestion is greatly appreciated, thanks!