If `x^(2)-x+1=0` then the value of `x^(3 )` is

Options

A. 1
B. -1
C. `-1,1`
D. 0

Answer

To solve the equation \( x^2 - x + 1 = 0 \) and find the value of \( x^3 \), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is \( x^2 - x + 1 = 0 \). Here, we can identify:
- \( a = 1 \)
- \( b = -1 \)
- \( c = 1 \)

### Step 2: Apply the quadratic formula
The quadratic formula to find the roots of the equation is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{1 \pm \sqrt{1 - 4}}{2} \]
\[ x = \frac{1 \pm \sqrt{-3}}{2} \]

### Step 3: Simplify the expression
Since \( \sqrt{-3} = i\sqrt{3} \), we can write:
\[ x = \frac{1 \pm i\sqrt{3}}{2} \]
Thus, the roots are:
\[ x_1 = \frac{1 + i\sqrt{3}}{2}, \quad x_2 = \frac{1 - i\sqrt{3}}{2} \]

### Step 4: Find \( x^3 \)
To find \( x^3 \), we can use the identity derived from the original equation. From \( x^2 - x + 1 = 0 \), we can express \( x^2 \) in terms of \( x \):
\[ x^2 = x - 1 \]
Now, to find \( x^3 \):
\[ x^3 = x \cdot x^2 = x(x - 1) = x^2 - x \]
Substituting \( x^2 \) from the previous step:
\[ x^3 = (x - 1) - x = -1 \]

### Conclusion
Thus, the value of \( x^3 \) is:
\[ \boxed{-1} \]