If `1 - I ` is a root of the equation `x^(2) + ax +b = 0 ` where` a b in R ` then value of a is

Options

A. `-2`
B. 2
C. 0
D. none of these

Answer

To solve the problem, we need to find the value of \( a \) given that \( 1 - i \) is a root of the quadratic equation \( x^2 + ax + b = 0 \).

### Step-by-Step Solution:

1. **Identify the roots**: Since \( 1 - i \) is a root of the equation and the coefficients \( a \) and \( b \) are real numbers, the complex roots must occur in conjugate pairs. Therefore, the other root will be \( 1 + i \).

2. **Sum of the roots**: The sum of the roots of a quadratic equation \( x^2 + ax + b = 0 \) is given by the formula:
\[ \text{Sum of roots} = -\frac{a}{1} \]
Since we have two roots \( 1 - i \) and \( 1 + i \), we can calculate their sum:
\[ (1 - i) + (1 + i) = 1 - i + 1 + i = 2 \]

3. **Set up the equation**: According to the formula for the sum of the roots:
\[ -a = 2 \]

4. **Solve for \( a \)**: Rearranging the equation gives:
\[ a = -2 \]

Thus, the value of \( a \) is \( -2 \).

### Final Answer:
\[ a = -2 \]