Fractional Equation!

Consider the function $f(x) = x^2 - x$ . Note that the function is strictly increasing and continuous for $x > 1$ .

If $\langle x^2 \rangle = \langle x \rangle$ , we must have $x^2 - x = n$ for some integer $n$ . In the range $[1, 7]$ , f(x) takes on every real value in $[0, 42]$ . It is also strictly increasing, so it takes on every real value in $[0, 42]$ exactly once.

Because there are 43 integers in $[0, 42]$ , there are 43 values of $x$ for which $x^2 - x = n$ , meaning that there are $\boxed{43}$ values of $x$ such that $\langle x^2 \rangle = \langle x \rangle$ .

As a bonus, solving the quadratic with the quadratic formula gives the values of $x$ as $x = \frac{1\pm{\sqrt{1+4n}}}{2}$ for $n$ in the range $[0, 42]$

It can be seen that $0 \le \langle x^2 \rangle < 1$ when $\sqrt{n} \le x < \sqrt{n+1}$ , where $n = 1,2,3, ...7$ and that $0 \le \langle x \rangle < 1$ when $n \le x < n+1$ . For $x \in [1,7]$ , there are $48$ ranges of $\langle x^2 \rangle \in [0,1)$ and $6$ ranges of $\langle x \rangle \in [0,1)$ . The graph above shows the curves of $\langle x^2 \rangle \in [0.1)$ and $\langle x \rangle \in [0.1)$ . We see that from $x \in [1,2)$ , $\langle x^2 \rangle \in [0.1)$ and $\langle x \rangle \in [0.1)$ meet at $x=1$ and $x \in (\sqrt{2},\sqrt{3})$ , and they do not meet in $x \in (\sqrt{3},2)$ as marked by the dash-line circle. Similarly, they do not meet in $x \in (\sqrt{8},3)$ , $\in (\sqrt{15},4)$ , $\in (\sqrt{24},5)$ , $\in (\sqrt{35},6)$ and $\in (\sqrt{48},7)$ .

Therefore, the number of solutions or the points $\langle x^2 \rangle$ and $\langle x \rangle$ meet $=48-6+1 = \boxed{43}$ .