If x is the root of the equation `x^(2) -ix -1 =0` , then
The value of ` x^(51)` is

Options

A. 1
B. `-1`
C. i
D. `-i`

Answer

To solve the equation \( x^2 - ix - 1 = 0 \) for the root \( x \) and subsequently find \( x^{51} \), we will follow these steps:

### Step 1: Solve the quadratic equation
The given equation is:
\[ x^2 - ix - 1 = 0 \]
We can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -i \), and \( c = -1 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (-i)^2 = -1 \]
\[ 4ac = 4 \cdot 1 \cdot (-1) = -4 \]
Thus, the discriminant is:
\[ b^2 - 4ac = -1 - (-4) = -1 + 4 = 3 \]

### Step 3: Substitute into the quadratic formula
Now substituting back into the quadratic formula:
\[ x = \frac{-(-i) \pm \sqrt{3}}{2 \cdot 1} = \frac{i \pm \sqrt{3}}{2} \]
So the roots are:
\[ x_1 = \frac{i + \sqrt{3}}{2}, \quad x_2 = \frac{i - \sqrt{3}}{2} \]

### Step 4: Find \( x^{51} \)
To find \( x^{51} \), we can use the properties of complex numbers. We will express \( x \) in polar form.

### Step 5: Calculate the modulus and argument
The modulus \( r \) of \( x_1 \) is:
\[ r = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1 \]
The argument \( \theta \) is:
\[ \theta = \tan^{-1}\left(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \]
Thus, we can express \( x_1 \) in polar form as:
\[ x_1 = e^{i\frac{\pi}{6}} \]

### Step 6: Use De Moivre's Theorem
Using De Moivre's theorem:
\[ x^{51} = \left(e^{i\frac{\pi}{6}}\right)^{51} = e^{i\frac{51\pi}{6}} = e^{i\left(8\pi + \frac{3\pi}{6}\right)} = e^{i\frac{3\pi}{6}} = e^{i\frac{\pi}{2}} \]

### Step 7: Simplify
Thus, we have:
\[ x^{51} = i \]

### Final Answer
The value of \( x^{51} \) is:
\[ \boxed{i} \]