Question:
If $[x]^{2}-5[x]+6=0$, where [.] denotes the greatest integer function, then
(a) x ∈ [3, 4]
(b) x ∈ (2, 3]
(c) x ∈ [2, 3]
(d) x ∈ [2, 4)
Solution:
The given equation is $[x]^{2}-5[x]+6=0$.
$[x]^{2}-5[x]+6=0$
$\Rightarrow[x]^{2}-3[x]-2[x]+6=0$
$\Rightarrow[x]([x]-3)-2([x]-3)=0$
$\Rightarrow([x]-2)([x]-3)=0$
$\Rightarrow[x]-2=0$ or $[x]-3=0$
$\Rightarrow[x]=2$ or $[x]=3$
$\Rightarrow x \in[2,3)$ or $x \in[3,4)$
$\Rightarrow x \in[2,4)$
Hence, the correct answer is option (d).
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.