If $[x]^2-5[x]+6=0$,where $[x]$ denotes the greatest integer less than or equal to $x$,then total set of values of $x$ is

If $[x]^2-5[x]+6=0$,where $[x]$ denotes the greatest integer less than or equal to $x$,then total set of values of $x$ is
$(A)x\in[3,4)$
$(B)x\in[2,3]$
$(C)x\in\left\{2,3\right\}$
$(D)x\in[2,4)$

My attempt:
$[x]^2-5[x]+6=0$
$[x]=2,[x]=3$
$2\leq x<3,3\leq x<4$

• Why not just start testing values? Dec 1, 2015 at 16:20
• Why downvote,would downvoter please comment? Dec 4, 2015 at 2:09
• The downvote is likely because this is a multiple choice question where you can find the correct answer just by guessing things, and you didn't find the answer. (When you plug in $x=2$ it satisfies the equation, so (A) is not the answer, if you plug in $x=2.5$ it satisfies the equation so (C) is not the answer, ...). This shows a concerning lack of effort on your part. Contrast this with what would have been a better question, "Here's this multiple choice question. I know the answer is _ by trial-and-error. How would you arrive at this answer in a more methodical way?" Dec 4, 2015 at 5:22

What you've done is correct, and now you have $$2\le x\lt 3\quad\color{red}{\text{or}}\quad 3\le x\lt 4.$$
$[x]=2 \Rightarrow x \in [2,3)$ or $[x]=3 \Rightarrow x \in [3,4)$
$\Rightarrow x \in [2,4)$
What you have't seen is $3$ is in one interval ad not in the other.