TITLE: Solve the equation $2x^2-[x]-1=0$ where $[x]$ is biggest integer not greater than $x$.
QUESTION [6 upvotes]: I' ve tried with $x^2 = {[x]+1\over 2}$ so $x$ is a square root of half integer. And know? What to do with that?

REPLY [2 votes]: Rewrite
$$x=\pm\sqrt{\frac{[x]+1}2}$$ and try increasing values for the integer $[x]$.

$[x]=-1\to x=0$: incompabible;
$[x]=0\to x=\pm\dfrac1{\sqrt 2}$: $x=\color{green}{\dfrac1{\sqrt2}}$ is a solution;
$[x]=1\to x=\pm1$: $x=\color{green}{1}$ is a solution;
$[x]=2\to x=\pm\sqrt{\dfrac32}$: from now on, the RHS is too small, no solution anymore.