If `[x]^(2)=[x+6]`, where [x]= the greatest integer less than or equal to x, then x must be such that

Options

A. x=3,-2
B. `x in [-2,-1)`
C. `x in [3,4)`
D. `x in [-2,-1) cup [3,4)`

Answer

To solve the equation \([x]^2 = [x + 6]\), where \([x]\) represents the greatest integer less than or equal to \(x\), we will follow these steps:

### Step 1: Understand the Greatest Integer Function
The greatest integer function, denoted as \([x]\), gives the largest integer less than or equal to \(x\). For example, \([3.7] = 3\) and \([-2.3] = -3\).

### Step 2: Rewrite the Equation
We start with the equation:
\[ [x]^2 = [x + 6] \]
Let \(n = [x]\). Then we can rewrite the equation as:
\[ n^2 = [x + 6] \]

### Step 3: Analyze \([x + 6]\)
Since \(n = [x]\), we know that \(n \leq x < n + 1\). Therefore, we can express \(x + 6\) as:
\[ n + 6 \leq x + 6 < n + 7 \]
This implies:
\[ [n + 6] = n + 6 \quad \text{(since \(n + 6\) is an integer)} \]

### Step 4: Set Up the Equation
Now we can substitute back into our equation:
\[ n^2 = n + 6 \]

### Step 5: Rearrange the Equation
Rearranging gives us:
\[ n^2 - n - 6 = 0 \]

### Step 6: Factor the Quadratic Equation
Now we will factor the quadratic:
\[ (n - 3)(n + 2) = 0 \]
This gives us two possible solutions:
\[ n - 3 = 0 \quad \Rightarrow \quad n = 3 \]
\[ n + 2 = 0 \quad \Rightarrow \quad n = -2 \]

### Step 7: Find the Corresponding \(x\) Values
1. For \(n = 3\):
\[ [x] = 3 \quad \Rightarrow \quad 3 \leq x < 4 \]

2. For \(n = -2\):
\[ [x] = -2 \quad \Rightarrow \quad -2 \leq x < -1 \]

### Step 8: Combine the Intervals
The values of \(x\) that satisfy the original equation are:
\[ x \in [-2, -1) \cup [3, 4) \]

### Final Answer
Thus, the solution set for \(x\) is:
\[ x \in [-2, -1) \cup [3, 4) \]

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