IF `(x-5)^3+ (x-6)^3 +(x-7)^3=3 (x-5) (x-6) (x-7)` then what is the value of x

Options

A. 7
B. 6
C. 5
D. 18

Answer

To solve the equation \((x-5)^3 + (x-6)^3 + (x-7)^3 = 3(x-5)(x-6)(x-7)\), we can use the identity for the sum of cubes. The identity states that:

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

In our case, we can let:
- \(a = x - 5\)
- \(b = x - 6\)
- \(c = x - 7\)

Now, substituting these into the equation, we have:

\[ (x-5)^3 + (x-6)^3 + (x-7)^3 - 3(x-5)(x-6)(x-7) = 0 \]

This can be rewritten using the identity:

\[ (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) = 0 \]

### Step 1: Calculate \(a + b + c\)

First, we calculate \(a + b + c\):

\[ a + b + c = (x - 5) + (x - 6) + (x - 7) = 3x - 18 \]

### Step 2: Set \(a + b + c = 0\)

Setting \(3x - 18 = 0\):

\[ 3x = 18 \]
\[ x = 6 \]

### Step 3: Verify the solution

Now we need to verify if \(x = 6\) satisfies the original equation.

Substituting \(x = 6\):

\[ (6-5)^3 + (6-6)^3 + (6-7)^3 = 1^3 + 0^3 + (-1)^3 = 1 + 0 - 1 = 0 \]

And for the right-hand side:

\[ 3(6-5)(6-6)(6-7) = 3(1)(0)(-1) = 0 \]

Both sides equal zero, confirming that \(x = 6\) is indeed a solution.

### Conclusion

Thus, the value of \(x\) is:

\[ \boxed{6} \]