While working through a question , I stumbled across the following equation

$$ c = \frac {(2-a\_0)+ \sqrt{-a\_0(3a\_0-4)}}{2} $$

We are required to maximise the function $c$ and find its corresponding value 

I plotted its graph on Desmos and found its maxima to be 1/3 however I have no clue how to arrive at this result

Also we are not allowed to use calculus (at least it is discouraged )
 How are we supposed to proceed with this?