Equation `3sqrt(-2(x+3))-1=|x+3|+a` has exactly two real roots , then the maximum possible value of |[a]| is _________. { where [.] denotes the greatest integer function } .

Answer

To solve the equation \( 3\sqrt{-2(x+3)} - 1 = |x+3| + a \) and find the maximum possible value of \( |[a]| \) where \([.]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Determine the domain of the square root
The expression under the square root must be non-negative:
\[ -2(x+3) \geq 0 \implies x + 3 \leq 0 \implies x \leq -3 \]
**Hint:** The square root function is defined only for non-negative values.

### Step 2: Rewrite the equation for \( x \leq -3 \)
Since \( x + 3 \) is negative for \( x \leq -3 \), we can express \( |x+3| \) as:
\[ |x+3| = -(x+3) = -x - 3 \]
Thus, the equation becomes:
\[ 3\sqrt{-2(x+3)} - 1 = -x - 3 + a \]

### Step 3: Rearranging the equation
Rearranging gives:
\[ 3\sqrt{-2(x+3)} = -x - 3 + a + 1 \]
\[ 3\sqrt{-2(x+3)} = -x + a - 2 \]

### Step 4: Square both sides
Squaring both sides to eliminate the square root:
\[ 9(-2(x+3)) = (-x + a - 2)^2 \]
\[ -18(x + 3) = (-x + a - 2)^2 \]

### Step 5: Expand the right-hand side
Expanding the right-hand side:
\[ -18x - 54 = x^2 - 2x(a - 2) + (a - 2)^2 \]
Rearranging gives:
\[ x^2 + (2a - 16)x + (a^2 - 4a + 4 + 54) = 0 \]
\[ x^2 + (2a - 16)x + (a^2 - 4a + 58) = 0 \]

### Step 6: Condition for exactly two real roots
For the quadratic equation to have exactly two real roots, the discriminant must be greater than zero:
\[ D = (2a - 16)^2 - 4(1)(a^2 - 4a + 58) > 0 \]
Calculating the discriminant:
\[ D = (2a - 16)^2 - 4(a^2 - 4a + 58) \]
Expanding:
\[ D = 4a^2 - 64a + 256 - 4a^2 + 16a - 232 > 0 \]
This simplifies to:
\[ -48a + 24 > 0 \implies 48a < 24 \implies a < \frac{1}{2} \]

### Step 7: Finding the maximum value of \( |[a]| \)
The maximum value of \( a \) is just below \( \frac{1}{2} \). Thus, the greatest integer function of \( a \) is:
\[ |[a]| = 0 \]
since \( a \) can be negative but is less than \( \frac{1}{2} \).

### Final Answer
The maximum possible value of \( |[a]| \) is:
\[ \boxed{0} \]