I have to find the equilibria of the following model: $$\frac{dI}{dt} = \frac{\alpha \beta I (1-I)}{\alpha I + r} - \mu I - \frac{\gamma I}{A+I}$$
and then find the conditions for their existence. For nontrivial equilibria, $f(I)$ gets a little messy and $f'(I)$ gets even messier. Before I undertake finding/classifying nontrivial roots using the quadratic equation, is there an easier way to find/classify the equilibria of this equation?
Ignoring the trivial solution $I=0$, you can simplify the equation to
$$ \frac{\beta (1-I)}{I + \frac{r}{\alpha}} - \mu - \frac{\gamma}{A+I} = 0 $$
Multiplying through by the common denominator
$$ \beta(1-I)(A+I) - \mu\left(I+\frac{r}{\alpha}\right)(A+I) - \gamma\left(I+\frac{r}{\alpha}\right) = 0 $$
It's just straight algebra from here to reduce this down to a quadratic equation.
If you don't feel like solving it, try finding the determinant to see how many roots it has.
Before you get deterred, remember that everything but $I$ is just a bunch of constants. Try reducing them to simpler constants if you can.