If [x] denotes the greatest integer function .Find the solution set of equation `[x]^2 +5[x] +6=0`

Answer

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

### Step 1: Substitute \(y = [x]\)
Let \(y = [x]\). The equation then becomes:
\[ y^2 + 5y + 6 = 0 \]

### Step 2: Factor the quadratic equation
Next, we will factor the quadratic equation:
\[ y^2 + 5y + 6 = (y + 2)(y + 3) = 0 \]

### Step 3: Find the roots
Setting each factor to zero gives us:
1. \(y + 2 = 0 \Rightarrow y = -2\)
2. \(y + 3 = 0 \Rightarrow y = -3\)

### Step 4: Determine the values of \(x\)
Since \(y = [x]\), we have two cases:
1. If \([x] = -2\), then \(-2 \leq x < -1\).
2. If \([x] = -3\), then \(-3 \leq x < -2\).

### Step 5: Combine the intervals
Now we combine the intervals obtained from both cases:
- From \([x] = -2\): \(-2 \leq x < -1\)
- From \([x] = -3\): \(-3 \leq x < -2\)

Thus, the complete solution set is:
\[ [-3, -2) \cup [-2, -1) \]

### Final Solution
The solution set of the equation \([x]^2 + 5[x] + 6 = 0\) is:
\[ [-3, -1) \]