If `( x - a) ( x - b) = 1 and a - b + 5 = 0 `, then the value of `( x - a)^(3)- (1)/((x - a)^(3))` is

Options

A. `-125`
B. 1
C. 125
D. 140

Answer

To solve the problem, we need to find the value of \( (x - a)^3 - \frac{1}{(x - a)^3} \) given that \( (x - a)(x - b) = 1 \) and \( a - b + 5 = 0 \).

### Step-by-Step Solution:

1. **Start with the given equations**:
\[ (x - a)(x - b) = 1 \]
\[ a - b + 5 = 0 \implies a - b = -5 \implies a = b - 5 \]

2. **Substituting \( a \) into the first equation**:
Substitute \( a = b - 5 \) into \( (x - a)(x - b) = 1 \):
\[ (x - (b - 5))(x - b) = 1 \]
Simplifying this gives:
\[ (x - b + 5)(x - b) = 1 \]

3. **Let \( y = x - b \)**:
Then we can rewrite the equation:
\[ (y + 5)y = 1 \]
Expanding this gives:
\[ y^2 + 5y - 1 = 0 \]

4. **Using the quadratic formula to solve for \( y \)**:
The quadratic formula is:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 5, c = -1 \):
\[ y = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{-5 \pm \sqrt{25 + 4}}{2} = \frac{-5 \pm \sqrt{29}}{2} \]

5. **Finding \( x - a \)**:
Since \( x - a = (x - b) + 5 = y + 5 \):
\[ x - a = \frac{-5 \pm \sqrt{29}}{2} + 5 = \frac{-5 \pm \sqrt{29} + 10}{2} = \frac{5 \pm \sqrt{29}}{2} \]

6. **Calculating \( (x - a)^3 - \frac{1}{(x - a)^3} \)**:
Let \( z = x - a \):
\[ z = \frac{5 \pm \sqrt{29}}{2} \]
We need to find \( z^3 - \frac{1}{z^3} \). We can use the identity:
\[ z^3 - \frac{1}{z^3} = (z - \frac{1}{z}) \left( z^2 + 1 + \frac{1}{z^2} \right) \]

7. **Finding \( z - \frac{1}{z} \)**:
Calculate \( z - \frac{1}{z} \):
\[ z - \frac{1}{z} = \frac{5 \pm \sqrt{29}}{2} - \frac{2}{5 \pm \sqrt{29}} = \frac{(5 \pm \sqrt{29})^2 - 4}{2(5 \pm \sqrt{29})} \]
Simplifying this gives:
\[ z - \frac{1}{z} = 5 \]

8. **Finding \( z^2 + 1 + \frac{1}{z^2} \)**:
Using \( z - \frac{1}{z} = 5 \):
\[ z^2 + \frac{1}{z^2} = (z - \frac{1}{z})^2 + 2 = 25 + 2 = 27 \]

9. **Putting it all together**:
Now, substituting back:
\[ z^3 - \frac{1}{z^3} = (z - \frac{1}{z})(z^2 + 1 + \frac{1}{z^2}) = 5 \cdot 27 = 135 \]

### Final Answer:
The value of \( (x - a)^3 - \frac{1}{(x - a)^3} \) is \( 135 \).