What is the result of dividing the complex number (3+2i) by (1-i)?

To divide the complex number \( \frac{3 + 2i}{1 - i} \), we use the method of multiplying both the numerator and the denominator by the conjugate of the denominator.

The conjugate of \( 1 - i \) is \( 1 + i \). Let's proceed step by step:

### Step 1: Multiply numerator and denominator by the conjugate of the denominator.
\[
\frac{3 + 2i}{1 - i} \times \frac{1 + i}{1 + i} = \frac{(3 + 2i)(1 + i)}{(1 - i)(1 + i)}
\]

### Step 2: Simplify the denominator.
\[
(1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 2
\]

### Step 3: Multiply the numerators.
\[
(3 + 2i)(1 + i) = 3(1) + 3(i) + 2i(1) + 2i(i)
\]
\[
= 3 + 3i + 2i + 2i^2 = 3 + 5i + 2(-1) = 3 + 5i - 2 = 1 + 5i
\]

### Step 4: Combine the results.
\[
\frac{1 + 5i}{2}
\]

### Step 5: Write the result in standard form.
\[
= \frac{1}{2} + \frac{5}{2}i
\]

Thus, the result of dividing \( 3 + 2i \) by \( 1 - i \) is:
\[
\frac{1}{2} + \frac{5}{2}i
\]

Would you like more details or explanations?

Here are 5 related questions:
1. How do you multiply complex numbers in polar form?
2. What is the significance of the conjugate in complex number division?
3. How do you convert a complex number from Cartesian to polar form?
4. Can you find the magnitude and argument of the complex number \( 3 + 2i \)?
5. How do you solve equations involving complex numbers?

**Tip**: Always simplify complex number operations by using conjugates to eliminate imaginary parts from denominators.

Discover how to divide the complex number (3+2i) by (1-i) using the conjugate method. This detailed step-by-step solution simplifies the process and highlights key concepts in complex numbers, making it perfect for Grades 10-12 students.

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