If one root of `x^(2)+x +1 = 0` is `(-1+sqrt(3)i)/(2)`, then other root is :

Options

A. A. `(-1 - sqrt(3)i)/(2)`
B. B. `(1-sqrt(3)i)/(2)`
C. C. `(1-sqrt(3)i^(2))/(2)`
D. D. `(1+sqrt(3)t)/(2)`

Answer

To find the other root of the quadratic equation \( x^2 + x + 1 = 0 \) given that one root is \( \alpha = \frac{-1 + \sqrt{3}i}{2} \), we can follow these steps:

### Step 1: Identify the given root
We are given one root of the equation:
\[ \alpha = \frac{-1 + \sqrt{3}i}{2} \]

### Step 2: Use the property of roots of quadratic equations
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} \]
In our case, \( a = 1 \) and \( b = 1 \), so:
\[ \alpha + \beta = -\frac{1}{1} = -1 \]

### Step 3: Substitute the known root into the sum of roots formula
Now, substituting the value of \( \alpha \) into the equation:
\[ \frac{-1 + \sqrt{3}i}{2} + \beta = -1 \]

### Step 4: Isolate \( \beta \)
To find \( \beta \), we rearrange the equation:
\[ \beta = -1 - \frac{-1 + \sqrt{3}i}{2} \]
This can be simplified as follows:
\[ \beta = -1 + \frac{1 - \sqrt{3}i}{2} \]
Now, we need to express \(-1\) in terms of a fraction with a denominator of 2:
\[ -1 = \frac{-2}{2} \]
Thus:
\[ \beta = \frac{-2 + 1 - \sqrt{3}i}{2} = \frac{-1 - \sqrt{3}i}{2} \]

### Step 5: Write the final result
Thus, the other root \( \beta \) is:
\[ \beta = \frac{-1 - \sqrt{3}i}{2} \]

### Conclusion
The other root of the equation \( x^2 + x + 1 = 0 \) is:
\[ \frac{-1 - \sqrt{3}i}{2} \]