What is the sum of the squares of the roots of the equation $x^2 − 7[x] + 5 = 0?$ (Here $[x]$ denotes the greatest integer less than or equal to $x$) I tried many different approaches to this question,
$α^2 = 7[α] - 5$ and 
$β^2 = 7[β] - 5$
So combining both the equations we get,
$α^2 + β^2 = 7([α] + [β]) - 10$,
Which is the answer. However I am unable to simplify the LHS.

$x^2–7[x]+5=0$
$\Rightarrow x^2=7[x]–5 \hspace{1cm}(1)$
$\Rightarrow x^2≤7x–5$ [Since $[x]≤x$]
$\Rightarrow x^2–7x+5≤0$
Therefore, $0.8≤x≤6.2$
i.e., $0≤[x]≤6$
Since for $[x]=0$, equation $(1)$ becomes, $x²=–5$, which is not possible.
Therefore, $1≤[x]≤6$.
In equation $(1)$
For $[x]=1, x^2=2 \Rightarrow x=\pm\sqrt{2}$
For $[x]=2, x²=9 \Rightarrow x=\pm\sqrt{3}$ [not possible because $x=\pm3$ and $[x]=2$ doesn’t make any sense]
For $[x]=3, x²=16 \Rightarrow x=\pm\sqrt{4}$ [not possible because $x=±4$ and $[x]=3$ doesn’t make any sense]
For $[x]=4, x²=23 \Rightarrow x=\pm\sqrt{23}$
For $[x]=5, x²=30 \Rightarrow x=\pm\sqrt{30}$
For $[x]=6, x²=37 \Rightarrow x=\pm\sqrt{37}$
Therefore, sum of all the roots of the given equation $=2+23+30+37=92$.
Hope it helps!!!!
