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Question
If \[\left[ x \right]^2 - 5\left[ x \right] + 6 = 0\], where [.] denotes the greatest integer function, then
Options
(a) x ∈ [3, 4]
(b) x ∈ (2, 3]
(c) x ∈ [2, 3]
(d) x ∈ [2, 4)
Solution
The given equation is \[\left[ x \right]^2 - 5\left[ x \right] + 6 = 0\]
\[\left[ x \right]^2 - 5\left[ x \right] + 6 = 0\]
\[ \Rightarrow \left[ x \right]^2 - 3\left[ x \right] - 2\left[ x \right] + 6 = 0\]
\[ \Rightarrow \left[ x \right]\left( \left[ x \right] - 3 \right) - 2\left( \left[ x \right] - 3 \right) = 0\]
\[ \Rightarrow \left( \left[ x \right] - 2 \right)\left( \left[ x \right] - 3 \right) = 0\]
\[\Rightarrow \left[ x \right] - 2 = 0 \text{ or } \left[ x \right] - 3 = 0\]
\[ \Rightarrow \left[ x \right] = 2\text{ or } \left[ x \right] = 3\]
⇒ x ∈ [2, 3) or x ∈ [3, 4)
⇒ x ∈ [2, 4)
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