Given `z_(1)=1 -i, z_(2)= -2 + 4i`, calculate the values of a and b if `a + bi= (z_(1)z_(2))/(z_(1))`

Answer

To solve the problem, we need to calculate the values of \( a \) and \( b \) from the expression \( a + bi = \frac{z_1 z_2}{z_1} \), where \( z_1 = 1 - i \) and \( z_2 = -2 + 4i \).

### Step-by-step Solution:

1. **Calculate \( z_1 z_2 \)**:
\[ z_1 z_2 = (1 - i)(-2 + 4i) \]
Using the distributive property (FOIL method):
\[ = 1 \cdot (-2) + 1 \cdot (4i) - i \cdot (-2) - i \cdot (4i) \]
\[ = -2 + 4i + 2i - 4i^2 \]
Since \( i^2 = -1 \):
\[ = -2 + 4i + 2i + 4 \]
\[ = 2 + 6i \]

2. **Calculate \( \frac{z_1 z_2}{z_1} \)**:
\[ \frac{z_1 z_2}{z_1} = \frac{2 + 6i}{1 - i} \]

3. **Rationalize the denominator**:
Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(2 + 6i)(1 + i)}{(1 - i)(1 + i)} \]
The denominator simplifies as follows:
\[ (1 - i)(1 + i) = 1^2 - (-1) = 1 + 1 = 2 \]
Now, calculate the numerator:
\[ (2 + 6i)(1 + i) = 2 \cdot 1 + 2 \cdot i + 6i \cdot 1 + 6i \cdot i \]
\[ = 2 + 2i + 6i - 6 \]
\[ = -4 + 8i \]

4. **Combine the results**:
\[ \frac{-4 + 8i}{2} = \frac{-4}{2} + \frac{8i}{2} = -2 + 4i \]

5. **Identify \( a \) and \( b \)**:
From the expression \( a + bi = -2 + 4i \), we can identify:
\[ a = -2, \quad b = 4 \]

### Final Result:
Thus, the values of \( a \) and \( b \) are:
\[ a = -2, \quad b = 4 \]