Question

If \([\cdot ]\) represents the greatest integer function of less than or equal to type then the solution set of the equation \([x]^{2}-5[x]+6=0\) is

• A. \([2,4)\)
• B. \([2,3)\)
• C. \([3,4)\)
• D. \(\{2,3\}\)

Hint:

Factorise the quadratic in [x] and convert the integer values of [x] into intervals for x.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Let \([x]=t\). The equation becomes a quadratic in \(t\).

Step 2: Solve for t
\(t^{2}-5t+6=0\Rightarrow (t-2)(t-3)=0\Rightarrow t=2\text{ or }3\).

Step 3: Convert to x
\([x]=2\Rightarrow 2\le x<3\) and \([x]=3\Rightarrow 3\le x<4\).

Step 4: Combine
The union is \(2\le x<4\), i.e. the interval \([2,4)\).

Final Answer:
The solution set is the interval [2, 4). \[ \boxed{\text{(A) }[2,4)} \]