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.

Note that there are more than two roots (the equation is not a quadratic polynomial). We solve the problem in
three steps.
1) For $x<1$, $x^2+5>0\geq 7[x]$. No roots in $(-\infty,1)$.
2) For $x\geq 7$, $x^2+5>7x\geq 7[x]$. No roots in $[7,+\infty)$.
3) Try to solve the equation for $x\in [1,7)$ where $[x]\in\{1,2,3,4,5,6\}$.
If $x\in[1,2)$ then $x^2=7[x]-5=2$ implies $x=\sqrt{2}\in [1,2)$. So $x=\sqrt{2}$ is a root.
If $x\in[2,3)$ then $x^2=7[x]-5=9$ implies $x=\sqrt{3}\not\in [2,3)$. No roots.
If $x\in[3,4)$ then $x^2=7[x]-5=16$ implies $x=4\not\in [3,4)$. No roots.
If $x\in[4,5)$ then $x^2=7[x]-5=23$ implies $x=\sqrt{23}\in [4,5)$.  So $x=\sqrt{23}$ is a root.
If $x\in[5,6)$ then $x^2=7[x]-5=30$ implies $x=\sqrt{30}\in [5,6)$.  So $x=\sqrt{30}$ is a root.
If $x\in[6,7)$ then $x^2=7[x]-5=37$ implies $x=\sqrt{37}\in [6,7)$.  So $x=\sqrt{37}$ is a root.
Finally the sum of the squares of the roots is $2+23+30+37=92$.
