Question

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

• A. $x \in [3, 4)$
• B. $x \in [2, 4)$
• C. $x \in [2, 3)$
• D. $x \in [2, 3]$

Answer 1

The Correct Option is B

Given Equation: \([x]^2 - 5[x] + 6 = 0\), where \([x]\) is the greatest integer function.

Step 1: Solve the quadratic equation for \([x]\).

Let \(y = [x]\). The equation becomes:

\(y^2 - 5y + 6 = 0\)

Factorizing: \((y - 2)(y - 3) = 0\)

Solutions: \(y = 2\) or \(y = 3\)

Step 2: Translate back to \([x]\).

Case 1: \([x] = 2\) ⇒ \(x \in [2, 3)\)

Case 2: \([x] = 3\) ⇒ \(x \in [3, 4)\)

Step 3: Combine the intervals.

The solution is \(x \in [2, 3) \cup [3, 4)\) = \([2, 4)\)

Option Analysis:

(A) \(x \in [3, 4]\) - Partial solution (misses [2,3))

(B) \(x \in [2, 4]\) - Includes all solutions (correct)

(C) \(x \in [2, 3]\) - Partial solution (misses [3,4))

(D) \(x \in (2, 3]\) - Incorrect boundaries

Final Answer: \(\boxed{B}\)

Answer 2

Let \( [x] \) denote the greatest integer less than or equal to \( x \). We are given the equation:

\[ [x]^2 - 5[x] + 6 = 0 \]

This is a quadratic equation in \( [x] \). We can factor it as follows:

\[ ([x] - 2)([x] - 3) = 0 \]

This gives two possible values for \( [x] \):

\[ [x] = 2 \quad \text{or} \quad [x] = 3 \]

If \( [x] = 2 \), then \( 2 \leq x < 3 \).
If \( [x] = 3 \), then \( 3 \leq x < 4 \).

Therefore, the solution for \( x \) is the interval \( 2 \leq x < 4 \). In interval notation, this is \( [2, 4) \).

Final Answer: The final answer is \( {[2,4)} \).