If ` x^(2) - 16 x + 59 = 0 ` then what is the value of ` ( x - 6) ^(2) + [(1)/(( x - 6) ^(2))]`

Options

A. (a) 14
B. (b) 18
C. (c) 16
D. (d) 20

Answer

To solve the equation \( x^2 - 16x + 59 = 0 \) and find the value of \( (x - 6)^2 + \frac{1}{(x - 6)^2} \), we will follow these steps:

### Step 1: Substitute \( t = x - 6 \)
We start by letting \( t = x - 6 \). This means that \( x = t + 6 \).

### Step 2: Substitute \( x \) in the original equation
Next, we substitute \( x \) in the equation \( x^2 - 16x + 59 = 0 \):
\[ (t + 6)^2 - 16(t + 6) + 59 = 0 \]

### Step 3: Expand the equation
Now, we expand the equation:
\[ (t^2 + 12t + 36) - (16t + 96) + 59 = 0 \]
This simplifies to:
\[ t^2 + 12t + 36 - 16t - 96 + 59 = 0 \]
Combining like terms gives:
\[ t^2 - 4t - 1 = 0 \]

### Step 4: Solve for \( t \) using the quadratic formula
We can solve the quadratic equation \( t^2 - 4t - 1 = 0 \) using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -4 \), and \( c = -1 \):
\[ t = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 + 4}}{2} = \frac{4 \pm \sqrt{20}}{2} = \frac{4 \pm 2\sqrt{5}}{2} = 2 \pm \sqrt{5} \]

### Step 5: Calculate \( (x - 6)^2 + \frac{1}{(x - 6)^2} \)
Now we need to find \( (x - 6)^2 + \frac{1}{(x - 6)^2} \), which is \( t^2 + \frac{1}{t^2} \).

### Step 6: Find \( t^2 + \frac{1}{t^2} \)
We know:
\[ t - \frac{1}{t} = 4 \]
Now, squaring both sides:
\[ \left(t - \frac{1}{t}\right)^2 = 16 \]
This expands to:
\[ t^2 - 2 + \frac{1}{t^2} = 16 \]
Rearranging gives:
\[ t^2 + \frac{1}{t^2} = 16 + 2 = 18 \]

### Final Answer
Thus, the value of \( (x - 6)^2 + \frac{1}{(x - 6)^2} \) is \( \boxed{18} \).