If ` x^(2) - 12 x + 33 = 0 ` ,t hen what is the value of ` ( x - 4) ^(2) + [ 1 // ( x - 4)^(2) ] `?

Options

A. 16
B. 14
C. 18
D. 20

Answer

To solve the equation \( x^2 - 12x + 33 = 0 \) and find the value of \( (x - 4)^2 + \frac{1}{(x - 4)^2} \), we can follow these steps:

### Step 1: Solve the quadratic equation
We start with the quadratic equation:
\[ x^2 - 12x + 33 = 0 \]
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -12, c = 33 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant:
\[ b^2 - 4ac = (-12)^2 - 4 \cdot 1 \cdot 33 = 144 - 132 = 12 \]

### Step 3: Find the roots
Now we can find the roots using the quadratic formula:
\[ x = \frac{12 \pm \sqrt{12}}{2} = \frac{12 \pm 2\sqrt{3}}{2} = 6 \pm \sqrt{3} \]
Thus, the roots are:
\[ x_1 = 6 + \sqrt{3}, \quad x_2 = 6 - \sqrt{3} \]

### Step 4: Substitute \( x - 4 \)
Next, we need to find \( (x - 4) \):
\[ x - 4 = (6 \pm \sqrt{3}) - 4 = 2 \pm \sqrt{3} \]

### Step 5: Calculate \( (x - 4)^2 \)
Now we calculate \( (x - 4)^2 \):
\[ (x - 4)^2 = (2 \pm \sqrt{3})^2 = 4 \pm 4\sqrt{3} + 3 = 7 \pm 4\sqrt{3} \]

### Step 6: Calculate \( \frac{1}{(x - 4)^2} \)
Next, we need to find \( \frac{1}{(x - 4)^2} \). Let's denote \( m = (x - 4)^2 \), then:
\[ m = 7 \pm 4\sqrt{3} \]
To find \( \frac{1}{m} \):
\[ \frac{1}{m} = \frac{1}{7 \pm 4\sqrt{3}} \]
We can rationalize the denominator:
\[ \frac{1}{m} = \frac{7 \mp 4\sqrt{3}}{(7 \pm 4\sqrt{3})(7 \mp 4\sqrt{3})} = \frac{7 \mp 4\sqrt{3}}{49 - 48} = 7 \mp 4\sqrt{3} \]

### Step 7: Calculate \( (x - 4)^2 + \frac{1}{(x - 4)^2} \)
Now we can find:
\[ (x - 4)^2 + \frac{1}{(x - 4)^2} = (7 \pm 4\sqrt{3}) + (7 \mp 4\sqrt{3}) = 14 \]

### Conclusion
Thus, the value of \( (x - 4)^2 + \frac{1}{(x - 4)^2} \) is:
\[ \boxed{14} \]