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↑ stuart clark:
![$x=[x]+\{x\}$](/mathtex/34/342df1f35dd3d6529493d72c7034feae.gif "kopírovat do textarea")
we can write ![$
[x^2]&=[x]+3\{x\}\\
[x^2]-[x]&=3\{x\}.
$](/mathtex/23/23b9a9c0c1fcb249a0a182d8d76d15f3.gif "kopírovat do textarea")
. Then 
is an integer and![$
[n^2]&=n+2\{n\}\\
n^2&=n\quad\Rightarrow\quad n=0,\ n=1\quad\Rightarrow\quad{\color{blue}x=0,\ x=1}.
$](/mathtex/c4/c42f5a2eda8d6929db2da8db3c9dd1f2.gif "kopírovat do textarea")
. Then 
where 
is an integer. Then![$
\biggl[n^2+\frac 23\,n+\frac 19\biggr]&=n+\frac 13+2\cdot\frac 13\\
\biggl[n^2+\frac 23\,n+\frac 19\biggr]&=n+1.
$](/mathtex/73/735f1cfbaa5b29c0fbfb053a3278a84d.gif "kopírovat do textarea")
![$x-1<[x]\leq x$](/mathtex/25/253284706db6db2b29fa6acdc9dc9198.gif "kopírovat do textarea")
, 
.![$
n^2+\frac 23\,n+\frac 19-1&<\biggl[n^2+\frac 23\,n+\frac 19\biggr] & &\wedge & \biggl[n^2+\frac 23\,n+\frac 19\biggr]&\leq n^2+\frac 23\,n+\frac 19\\
n^2+\frac 23\,n-\frac 89&<n+1 & &\wedge & n+1&\leq n^2+\frac 23\,n+\frac 19\\
n^2-\frac 13\,n-\frac{17}9&<0 & &\wedge & 0&\leq n^2-\frac 13\,n-\frac 89\\
n&\in\{-1,0,1\} & &\wedge & n&\in\mathbb{Z}\setminus\{0,1\}.
$](/mathtex/42/42774c5da5b7d001a6a093e64723c3a6.gif "kopírovat do textarea")
.
. Then 
where ![$
\biggl[n^2+\frac 43\,n+\frac 49\biggr]&=n+\frac 23+2\cdot\frac 23\\
\biggl[n^2+\frac 43\,n+\frac 49\biggr]&=n+2.
$](/mathtex/db/db59d64322a47b4901de1a6d27e8b82d.gif "kopírovat do textarea")
![$
n^2+\frac 43\,n+\frac 49-1&<\biggl[n^2+\frac 43\,n+\frac 49\biggr] & &\wedge & \biggl[n^2+\frac 43\,n+\frac 49\biggr]&\leq n^2+\frac 43\,n+\frac 49\\
n^2+\frac 43\,n-\frac 59&<n+2 & &\wedge & n+2&\leq n^2+\frac 43\,n+\frac 49\\
n^2+\frac 13\,n-\frac{23}9&<0 & &\wedge & 0&\leq n^2+\frac 13\,n-\frac {14}9\\
n&\in\{-1,0,1\} & &\wedge & n&\in\mathbb{Z}\setminus\{-1,0,1\}.
$](/mathtex/eb/eb251c064657597dd10732ce0566bb46.gif "kopírovat do textarea")
.