The equation `x^3-3x+[a]=0,` where [*] denotes the greatest integer function, will have three real and distinct roots then range of a is

Answer

To solve the problem, we need to determine the range of \( a \) such that the equation \( x^3 - 3x + [a] = 0 \) has three real and distinct roots. Here, \([a]\) denotes the greatest integer function.

### Step-by-Step Solution:

1. **Define the function**:
Let \( f(x) = x^3 - 3x + [a] \).

2. **Differentiate the function**:
We find the derivative:
\[ f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x - 1)(x + 1) \]
Setting the derivative equal to zero gives us the critical points:
\[ f'(x) = 0 \implies x = -1, 1 \]

3. **Evaluate the function at the critical points**:
We need to check the values of \( f(-1) \) and \( f(1) \) to ensure that the function changes signs around these points.

- **Calculate \( f(-1) \)**:
\[ f(-1) = (-1)^3 - 3(-1) + [a] = -1 + 3 + [a] = 2 + [a] \]
For \( f(-1) > 0 \):
\[ 2 + [a] > 0 \implies [a] > -2 \]

- **Calculate \( f(1) \)**:
\[ f(1) = (1)^3 - 3(1) + [a] = 1 - 3 + [a] = -2 + [a] \]
For \( f(1) < 0 \):
\[ -2 + [a] < 0 \implies [a] < 2 \]

4. **Combine the inequalities**:
From the two conditions we derived:
\[ -2 < [a] < 2 \]
This means that \([a]\) can take integer values of \(-1, 0, 1\).

5. **Determine the range of \( a \)**:
- If \([a] = -1\), then \( -1 \leq a < 0 \).
- If \([a] = 0\), then \( 0 \leq a < 1 \).
- If \([a] = 1\), then \( 1 \leq a < 2 \).

Therefore, the combined range of \( a \) is:
\[ -1 \leq a < 2 \]

### Final Answer:
The range of \( a \) such that the equation \( x^3 - 3x + [a] = 0 \) has three real and distinct roots is:
\[ [-1, 2) \]