Solve `[x]^(2)-5[x]+6=0.`

Explanation

To solve the equation \([x]^2 - 5[x] + 6 = 0\), where \([x]\) denotes the greatest integer function (also known as the floor function), we can follow these steps:

### Step 1: Factor the quadratic equation
We start with the equation:
\[ [x]^2 - 5[x] + 6 = 0 \]
We can factor this quadratic equation. We need to find two numbers that multiply to \(6\) (the constant term) and add up to \(-5\) (the coefficient of the linear term). The numbers \(-2\) and \(-3\) fit this requirement.

Thus, we can factor the equation as:
\[ ([x] - 2)([x] - 3) = 0 \]

### Step 2: Solve for \([x]\)
Setting each factor equal to zero gives us:
1. \([x] - 2 = 0 \Rightarrow [x] = 2\)
2. \([x] - 3 = 0 \Rightarrow [x] = 3\)

### Step 3: Determine the values of \(x\)
Now we need to find the corresponding values of \(x\) for each case:

- For \([x] = 2\):
The greatest integer function \([x] = 2\) implies that \(x\) is in the interval:
\[ 2 \leq x < 3 \]

- For \([x] = 3\):
The greatest integer function \([x] = 3\) implies that \(x\) is in the interval:
\[ 3 \leq x < 4 \]

### Step 4: Combine the intervals
Now we combine the intervals we found:
- From \([x] = 2\), we have \(x \in [2, 3)\).
- From \([x] = 3\), we have \(x \in [3, 4)\).

Thus, the solution set for \(x\) is:
\[ x \in [2, 3) \cup [3, 4) \]

### Final Answer
The solution to the equation \([x]^2 - 5[x] + 6 = 0\) is:
\[ x \in [2, 4) \]