I have been thinking about solving equations involving several groups of floor functions. My research on solving floor function equations has only shown sums of individual floors, such as:
$\lfloor f(x) \rfloor + \lfloor g(x) \rfloor + \lfloor h(x) \rfloor = \lfloor i(x) \rfloor$
However, I have not found an example in which there are floors multiplied together, such as:
$\lfloor f(x) \rfloor \lfloor g(x) \rfloor + \lfloor h(x) \rfloor \lfloor i(x) \rfloor = 0$
I cannot figure out how to solve such a problem. As an example, I have the equation:
$\Bigl\lfloor \sqrt{2x + \frac{1}{2}} \Bigr \rfloor \lfloor x \rfloor - \frac{1}{2} \Bigl \lfloor \frac{\sqrt{8x} - 1}{2} \Bigr \rfloor^2 \Bigl \lfloor \sqrt{2x + \frac{1}{2}} \Bigr \rfloor - \frac{1}{2} \left( \frac{\sqrt{8x} - 1}{2} \right) \left( \sqrt{2x + \frac{1}{2}} \right) - P = 0$
I have attempted to break this into pieces, and set them equal to each other, however, the solutions which I have found do not seem to relate to one another, and do not give me an overall solution. What should I do?