If `[x]^(2)=[x+2]`, where [x]=the greatest integer integer less than or equal to x, then x must be such that

Options

A. x=2,-1
B. `x in [2,3)`
C. `x in [1,0)`
D. None of these

Explanation

To solve the equation \([x]^2 = [x + 2]\), where \([x]\) denotes the greatest integer less than or equal to \(x\), we can follow these steps:

### Step 1: Understand the greatest integer function
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). For example, if \(x = 2.3\), then \([x] = 2\).

### Step 2: Rewrite the equation
We can rewrite the equation as:
\[ [x]^2 = [x + 2] \]
This means that the square of the greatest integer less than or equal to \(x\) is equal to the greatest integer less than or equal to \(x + 2\).

### Step 3: Analyze the right-hand side
The expression \([x + 2]\) can be rewritten in terms of \([x]\):
\[ [x + 2] = [x] + 2 \quad \text{(if \([x]\) is an integer)} \]
This holds because adding 2 to \(x\) will increase the greatest integer by 2, provided \(x\) is not an integer.

### Step 4: Set up the equation
Now we can set up the equation:
\[ [x]^2 = [x] + 2 \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ [x]^2 - [x] - 2 = 0 \]

### Step 6: Factor the quadratic equation
This quadratic can be factored as:
\[ ([x] - 2)([x] + 1) = 0 \]
Thus, the solutions for \([x]\) are:
\[ [x] = 2 \quad \text{or} \quad [x] = -1 \]

### Step 7: Find the corresponding ranges for \(x\)
1. If \([x] = 2\):
- This means \(2 \leq x < 3\).

2. If \([x] = -1\):
- This means \(-1 \leq x < 0\).

### Step 8: Combine the ranges
The combined ranges of \(x\) are:
\[ [-1, 0) \cup [2, 3) \]

### Conclusion
Thus, the values of \(x\) that satisfy the equation \([x]^2 = [x + 2]\) are in the intervals:
\[ [-1, 0) \cup [2, 3) \]