Let [t] denote the greatest integer st. Then the equation in `x,[x]^2+2[x+2]-7=0` has :

Options

A. infinitely many solutions.
B. exactly four integral solutions.
C. no integral solution.
D. exactly two solutions .

Answer

To solve the equation \([x]^2 + 2[x + 2] - 7 = 0\), where \([x]\) denotes the greatest integer function (also known as the floor function), we can follow these steps:

### Step 1: Rewrite the Equation
Let \( t = [x] \). Then, the equation can be rewritten as:
\[ t^2 + 2(t + 2) - 7 = 0 \]

### Step 2: Simplify the Equation
Expanding the equation gives:
\[ t^2 + 2t + 4 - 7 = 0 \]
This simplifies to:
\[ t^2 + 2t - 3 = 0 \]

### Step 3: Factor the Quadratic Equation
Next, we can factor the quadratic equation:
\[ t^2 + 3t - t - 3 = 0 \]
This can be factored as:
\[ (t + 3)(t - 1) = 0 \]

### Step 4: Solve for \( t \)
Setting each factor to zero gives us:
1. \( t + 3 = 0 \) → \( t = -3 \)
2. \( t - 1 = 0 \) → \( t = 1 \)

### Step 5: Determine the Values of \( x \)
Since \( t = [x] \), we can find the corresponding intervals for \( x \):
- For \( t = -3 \):
\[ -3 \leq x < -2 \]
- For \( t = 1 \):
\[ 1 \leq x < 2 \]

### Step 6: Identify the Solutions
The intervals for \( x \) are:
1. From \( -3 \) to \( -2 \) (including -3 and excluding -2)
2. From \( 1 \) to \( 2 \) (including 1 and excluding 2)

### Step 7: Count the Solutions
- The interval \( [-3, -2) \) contains infinitely many values.
- The interval \( [1, 2) \) also contains infinitely many values.

Thus, the equation has infinitely many solutions.

### Final Answer
The equation has infinitely many solutions.

---