Question

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

(a) x ∈ [3, 4] (b) x ∈ (2, 3] (c) x ∈ [2, 3] (d) x ∈ [2, 4)

Answer 1

The given equation is ${\left[x\right]}^2-5\left[x\right]+6=0$.

$$\begin{gathered} {\left[x\right]}^2-5\left[x\right]+6=0 \\ \Rightarrow {\left[x\right]}^2-3\left[x\right]-2\left[x\right]+6=0 \\ \Rightarrow \left[x\right]\left(\left[x\right]-3\right)-2\left(\left[x\right]-3\right)=0 \\ \Rightarrow \left(\left[x\right]-2\right)\left(\left[x\right]-3\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow \left[x\right]-2=0\mathrm{or}\left[x\right]-3=0 \\ \Rightarrow \left[x\right]=2\mathrm{or}\left[x\right]=3 \end{gathered}$$
⇒ x ∈ [2, 3) or x ∈ [3, 4)
⇒ x ∈ [2, 4)

Hence, the correct answer is option (d).