If for non - zero ` x, x^(2) - 4x - 1 = 0` the value of ` x^(2) + (1)/( x^(2))` is

Options

A. A)4
B. B)10
C. C)12
D. D)18

Answer

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

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

### Step 2: Calculate the discriminant
First, we calculate the discriminant:
\[ D = b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot (-1) = 16 + 4 = 20 \]

### Step 3: Find the roots using the quadratic formula
Now, we can find the roots:
\[ x = \frac{4 \pm \sqrt{20}}{2} = \frac{4 \pm 2\sqrt{5}}{2} = 2 \pm \sqrt{5} \]
Thus, the roots are \( x = 2 + \sqrt{5} \) and \( x = 2 - \sqrt{5} \).

### Step 4: Find \( x^2 \)
Next, we need to find \( x^2 \). We can use the fact that \( x^2 = 4x + 1 \) from rearranging the original equation:
\[ x^2 = 4x + 1 \]

### Step 5: Find \( \frac{1}{x} \)
To find \( \frac{1}{x} \), we can take the reciprocal of \( x \):
\[ \frac{1}{x} = \frac{1}{2 \pm \sqrt{5}} \]
To rationalize this, we multiply the numerator and denominator by the conjugate:
\[ \frac{1}{x} = \frac{2 \mp \sqrt{5}}{(2 \pm \sqrt{5})(2 \mp \sqrt{5})} = \frac{2 \mp \sqrt{5}}{4 - 5} = 2 \mp \sqrt{5} \]

### Step 6: Find \( \frac{1}{x^2} \)
Now, we can find \( \frac{1}{x^2} \):
\[ \frac{1}{x^2} = \left(\frac{1}{x}\right)^2 = (2 \mp \sqrt{5})^2 = 4 \mp 4\sqrt{5} + 5 = 9 \mp 4\sqrt{5} \]

### Step 7: Calculate \( x^2 + \frac{1}{x^2} \)
Now we can find \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = (4x + 1) + (9 \mp 4\sqrt{5}) = 4x + 10 \mp 4\sqrt{5} \]
However, we can simplify this further using the identity:
\[ x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2 \]

### Step 8: Find \( x + \frac{1}{x} \)
We can find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = (2 + \sqrt{5}) + (2 - \sqrt{5}) = 4 \]

### Step 9: Calculate \( (x + \frac{1}{x})^2 \)
Now we square it:
\[ (x + \frac{1}{x})^2 = 4^2 = 16 \]

### Step 10: Final calculation
Finally, we find:
\[ x^2 + \frac{1}{x^2} = 16 - 2 = 14 \]

Thus, the value of \( x^2 + \frac{1}{x^2} \) is \( 14 \).