If one root of the equation ` i x ^2 - 2(1+i) x+ (2-i) =0` is 2- I then the root is

Options

A. 2 + i
B. 1-2i
C. 1
D. -i

Answer

To solve the quadratic equation \( i x^2 - 2(1+i)x + (2-i) = 0 \) given that one root is \( 2 - i \), we will find the other root step by step.

### Step 1: Identify coefficients
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). From the given equation, we can identify:
- \( a = i \)
- \( b = -2(1+i) = -2 - 2i \)
- \( c = 2 - i \)

**Hint:** Compare the given equation with the standard form to identify \( a \), \( b \), and \( c \).

### Step 2: Calculate the sum of the roots
The sum of the roots of a quadratic equation is given by the formula:
\[ \text{Sum of roots} = -\frac{b}{a} \]
Substituting the values of \( b \) and \( a \):
\[ \text{Sum of roots} = -\frac{-2(1+i)}{i} = \frac{2(1+i)}{i} \]

**Hint:** Remember that dividing by \( i \) can be simplified by multiplying the numerator and denominator by \( -i \).

### Step 3: Simplify the sum of the roots
To simplify \( \frac{2(1+i)}{i} \):
\[ \frac{2(1+i)}{i} = 2(1+i) \cdot \frac{-i}{-i} = 2(-i - 1) = -2i - 2 \]
Thus, the sum of the roots is:
\[ -2 - 2i \]

**Hint:** Use the property of complex numbers to simplify expressions involving \( i \).

### Step 4: Use the known root to find the other root
Let the other root be \( y \). Since we know one root is \( 2 - i \), we can set up the equation:
\[ (2 - i) + y = -2 - 2i \]
Now, solving for \( y \):
\[ y = -2 - 2i - (2 - i) = -2 - 2i - 2 + i = -4 - i \]

**Hint:** Isolate \( y \) by moving the known root to the other side of the equation.

### Final Step: Conclusion
Thus, the other root of the equation is:
\[ \boxed{-4 - i} \]