If [x] denotes the greatest integer `le x` , then number of solutions of the equation `x^(2) - 2 - 2 [x] =0 ` is

Options

A. 4
B. 2
C. 3
D. none of these

Answer

To solve the equation \( x^2 - 2 - 2[x] = 0 \), where \([x]\) denotes the greatest integer less than or equal to \(x\), we can follow these steps:

### Step 1: Rewrite the equation
We can rearrange the equation to isolate the greatest integer function:
\[ x^2 - 2 = 2[x] \]

### Step 2: Analyze the left-hand side
The left-hand side, \(x^2 - 2\), is a quadratic function that opens upwards. It has roots at \(x = -\sqrt{2}\) and \(x = \sqrt{2}\). The vertex of this parabola is at \(x = 0\), where \(x^2 - 2 = -2\).

### Step 3: Determine the range of \([x]\)
The right-hand side, \(2[x]\), is a step function that takes integer values multiplied by 2. The value of \([x]\) will depend on the interval in which \(x\) lies:
- If \(n \leq x < n+1\), then \([x] = n\).

### Step 4: Set intervals based on \([x]\)
We will consider different intervals for \(x\) based on integer values of \([x]\):

1. **For \([x] = 0\)** (i.e., \(0 \leq x < 1\)):
\[ x^2 - 2 = 0 \implies x^2 = 2 \implies x = \sqrt{2} \text{ (not in the interval)} \]

2. **For \([x] = 1\)** (i.e., \(1 \leq x < 2\)):
\[ x^2 - 2 = 2 \implies x^2 = 4 \implies x = 2 \text{ (not in the interval)} \]

3. **For \([x] = 2\)** (i.e., \(2 \leq x < 3\)):
\[ x^2 - 2 = 4 \implies x^2 = 6 \implies x = \sqrt{6} \text{ (approximately 2.45, in the interval)} \]

4. **For \([x] = 3\)** (i.e., \(3 \leq x < 4\)):
\[ x^2 - 2 = 6 \implies x^2 = 8 \implies x = 2\sqrt{2} \text{ (approximately 2.83, not in the interval)} \]

5. **For \([x] = -1\)** (i.e., \(-1 \leq x < 0\)):
\[ x^2 - 2 = -2 \implies x^2 = 0 \implies x = 0 \text{ (not in the interval)} \]

6. **For \([x] = -2\)** (i.e., \(-2 \leq x < -1\)):
\[ x^2 - 2 = -4 \implies x^2 = -2 \text{ (no real solutions)} \]

### Step 5: Count the solutions
From our analysis:
- The only valid solution occurs in the interval where \([x] = 2\), giving us one solution: \(x = \sqrt{6}\).

### Conclusion
The total number of solutions to the equation \(x^2 - 2 - 2[x] = 0\) is **1**.

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