If x satisfies the cubic equation `ax^(3)+bx^(2)+cx+d=0` such that `cos^(-1)(x)+cos^(-1)(2x)+cos^(-1)(3x)=pi`, then find the value of `(b+c)-(a+d)`.

Answer

To solve the problem, we need to analyze the given equation and find the values of \( a, b, c, \) and \( d \) from the cubic equation \( ax^3 + bx^2 + cx + d = 0 \) based on the condition \( \cos^{-1}(x) + \cos^{-1}(2x) + \cos^{-1}(3x) = \pi \).

### Step-by-Step Solution

1. **Start with the given equation:**
\[ \cos^{-1}(x) + \cos^{-1}(2x) + \cos^{-1}(3x) = \pi \]

2. **Rearrange the equation:**
\[ \cos^{-1}(2x) + \cos^{-1}(3x) = \pi - \cos^{-1}(x) \]

3. **Use the property of cosine inverse:**
From the identity \( \cos^{-1}(a) + \cos^{-1}(b) = \cos^{-1}(ab - \sqrt{(1-a^2)(1-b^2)}) \), we can set:
\[ \cos^{-1}(2x) + \cos^{-1}(3x) = \cos^{-1}(2x \cdot 3x - \sqrt{(1 - (2x)^2)(1 - (3x)^2)}) \]

4. **Substituting values:**
\[ \cos^{-1}(2x) + \cos^{-1}(3x) = \cos^{-1}(6x^2 - \sqrt{(1 - 4x^2)(1 - 9x^2)}) \]

5. **Equating the two expressions:**
\[ \pi - \cos^{-1}(x) = \cos^{-1}(6x^2 - \sqrt{(1 - 4x^2)(1 - 9x^2)}) \]

6. **Taking cosine of both sides:**
\[ x = 6x^2 - \sqrt{(1 - 4x^2)(1 - 9x^2)} \]

7. **Rearranging the equation:**
\[ 6x^2 - x - \sqrt{(1 - 4x^2)(1 - 9x^2)} = 0 \]

8. **Square both sides to eliminate the square root:**
\[ (6x^2 - x)^2 = (1 - 4x^2)(1 - 9x^2) \]

9. **Expanding both sides:**
- Left side:
\[ (6x^2 - x)^2 = 36x^4 - 12x^3 + x^2 \]
- Right side:
\[ (1 - 4x^2)(1 - 9x^2) = 1 - 13x^2 + 36x^4 \]

10. **Setting both sides equal:**
\[ 36x^4 - 12x^3 + x^2 = 1 - 13x^2 + 36x^4 \]

11. **Cancelling \( 36x^4 \) from both sides:**
\[ -12x^3 + x^2 + 13x^2 - 1 = 0 \]
\[ -12x^3 + 14x^2 - 1 = 0 \]

12. **Rearranging gives:**
\[ 12x^3 - 14x^2 + 1 = 0 \]

13. **Identifying coefficients:**
From the equation \( 12x^3 - 14x^2 + 0x + 1 = 0 \), we have:
- \( a = 12 \)
- \( b = -14 \)
- \( c = 0 \)
- \( d = 1 \)

14. **Calculating \( (b + c) - (a + d) \):**
\[ (b + c) - (a + d) = (-14 + 0) - (12 + 1) = -14 - 13 = -27 \]

### Final Answer
The value of \( (b + c) - (a + d) \) is \( -27 \).