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: Rewrite the Equation
We start with the equation:
\[ [x]^2 - 5[x] + 6 = 0 \]

### Step 2: Factor the Quadratic
Next, we can factor the quadratic expression. We need two numbers that multiply to \(6\) (the constant term) and add up to \(-5\) (the coefficient of the linear term). The numbers \(-2\) and \(-3\) satisfy this condition. Thus, we can factor the equation as:
\[ ([x] - 2)([x] - 3) = 0 \]

### Step 3: Set Each Factor to Zero
Now, we set each factor equal to zero:
1. \([x] - 2 = 0\) which gives \([x] = 2\)
2. \([x] - 3 = 0\) which gives \([x] = 3\)

### Step 4: Determine the Values of \(x\)
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). Therefore, we can find the ranges for \(x\) based on the values of \([x]\):
1. If \([x] = 2\), then:
\[ 2 \leq x < 3 \]
2. If \([x] = 3\), then:
\[ 3 \leq x < 4 \]

### Step 5: Combine the Results
Combining the results from both cases, we find:
- For \([x] = 2\), \(x\) can take any value in the interval \([2, 3)\).
- For \([x] = 3\), \(x\) can take any value in the interval \([3, 4)\).

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

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