We have this equation over real numbers: $ \lfloor x^2 \rfloor = x+6$ . How we can solve it without guessing the answers?
My try: I tried to solve it in several intervals and found $-2$ and $3$ as answers but I can't deduct that there isn't any solution in other intervals .
Since LHS is a integer that implies $x+6$ is a integer which then implies that $x$ is a integer.Since $x\in\Bbb{Z}$ we know that $\lfloor x^2\rfloor=x^2$ Hence we have that $$x^2-x-6=0\\x_{1,2}=-2,3$$ Just a note if one of the solutions wasn't a integer then that soulution wouldn't satisfy the equation.