If
Question:

If $[x]^{2}-5[x]+6=0$, where [.] denotes the greatest integer function, then

(a) ∈ [3, 4]                           

(b) ∈ (2, 3]                           

(c) ∈ [2, 3]                           

(d) ∈ [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).

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