Let `px^(2)+qx+r=0` be a quadratic equation `(p,q,r in R)` such that its roots are `alpha and beta`. If `p+q+rlt0,p-q+rlt0 and r gt0`, then the value of `[alpha]+[beta]` is (where[x] denotes the greatest integer x)________.

Explanation

To solve the problem step by step, we will analyze the quadratic equation \( px^2 + qx + r = 0 \) and the conditions given.

### Step 1: Understanding the conditions
We have the following conditions:
1. \( p + q + r < 0 \)
2. \( p - q + r < 0 \)
3. \( r > 0 \)

### Step 2: Evaluate the function at specific points
We will evaluate the quadratic function \( f(x) = px^2 + qx + r \) at \( x = 0 \), \( x = 1 \), and \( x = -1 \).

- **At \( x = 0 \)**:
\[ f(0) = r > 0 \]
This means the function is positive at \( x = 0 \).

- **At \( x = 1 \)**:
\[ f(1) = p + q + r < 0 \]
This means the function is negative at \( x = 1 \).

- **At \( x = -1 \)**:
\[ f(-1) = p - q + r < 0 \]
This means the function is negative at \( x = -1 \).

### Step 3: Analyzing the intervals for roots
From the evaluations:
- Since \( f(0) > 0 \) and \( f(1) < 0 \), there is at least one root in the interval \( (0, 1) \).
- Since \( f(-1) < 0 \) and \( f(0) > 0 \), there is at least one root in the interval \( (-1, 0) \).

Let’s denote the roots as \( \alpha \) and \( \beta \):
- One root \( \alpha \) lies in \( (-1, 0) \).
- The other root \( \beta \) lies in \( (0, 1) \).

### Step 4: Finding the greatest integer values
- The greatest integer less than \( \alpha \) (which is between -1 and 0) is:
\[ [\alpha] = -1 \]
- The greatest integer less than \( \beta \) (which is between 0 and 1) is:
\[ [\beta] = 0 \]

### Step 5: Calculating the final result
Now, we need to find:
\[ [\alpha] + [\beta] = -1 + 0 = -1 \]

Thus, the value of \( [\alpha] + [\beta] \) is \( \boxed{-1} \).