If `(x-4)^(3) + ( x-5)^(3) +(x-3)^(3)=3(x-4) (x-5)(x-3)`. Then what is the value of x ?

Options

A. 18
B. 6
C. 7
D. 4

Answer

To solve the equation \((x-4)^{3} + (x-5)^{3} + (x-3)^{3} = 3(x-4)(x-5)(x-3)\), we can use the identity for the sum of cubes.

### Step-by-Step Solution:

1. **Identify the terms**: Let \(a = x - 4\), \(b = x - 5\), and \(c = x - 3\). The equation can be rewritten as:
\[ a^3 + b^3 + c^3 = 3abc \]

2. **Use the identity**: We know from algebra that:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
If \(a + b + c = 0\), then:
\[ a^3 + b^3 + c^3 = 3abc \]

3. **Calculate \(a + b + c\)**:
\[ a + b + c = (x - 4) + (x - 5) + (x - 3) = 3x - 12 \]

4. **Set the sum to zero**:
\[ 3x - 12 = 0 \]

5. **Solve for \(x\)**:
\[ 3x = 12 \implies x = \frac{12}{3} = 4 \]

6. **Conclusion**: The value of \(x\) is \(4\).