If `a-3 - (1)/(a - 3) = 5,` then the avlue of `(a- 3) ^(3) - (1)/((a - 3) ^(3))` is.

Options

A. 5
B. 7
C. 2
D. 14

Answer

To solve the equation \( a - 3 - \frac{1}{a - 3} = 5 \) and find the value of \( (a - 3)^3 - \frac{1}{(a - 3)^3} \), we can follow these steps:

### Step 1: Let \( x = a - 3 \)
We start by substituting \( x \) for \( a - 3 \). This simplifies our equation to:
\[ x - \frac{1}{x} = 5 \]

### Step 2: Rearranging the equation
To eliminate the fraction, we can multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 - 1 = 5x \]

### Step 3: Rearranging into standard form
Rearranging gives us a standard quadratic equation:
\[ x^2 - 5x - 1 = 0 \]

### Step 4: Using the quadratic formula
We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -5, c = -1 \):
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ x = \frac{5 \pm \sqrt{25 + 4}}{2} \]
\[ x = \frac{5 \pm \sqrt{29}}{2} \]

### Step 5: Finding \( (a - 3)^3 - \frac{1}{(a - 3)^3} \)
Now we need to compute \( x^3 - \frac{1}{x^3} \). We can use the identity:
\[ x^3 - \frac{1}{x^3} = \left(x - \frac{1}{x}\right)\left(x^2 + 1 + \frac{1}{x^2}\right) \]
From Step 1, we know \( x - \frac{1}{x} = 5 \).

### Step 6: Finding \( x^2 + \frac{1}{x^2} \)
To find \( x^2 + \frac{1}{x^2} \), we can use the identity:
\[ x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2 \]
Calculating this gives:
\[ x^2 + \frac{1}{x^2} = 5^2 + 2 = 25 + 2 = 27 \]

### Step 7: Substitute back into the identity
Now substituting back into our expression for \( x^3 - \frac{1}{x^3} \):
\[ x^3 - \frac{1}{x^3} = 5 \cdot (27) = 135 \]

### Final Answer
Thus, the value of \( (a - 3)^3 - \frac{1}{(a - 3)^3} \) is:
\[ \boxed{135} \]