To simplify \(\frac{6 + 3i}{4 - 2i}\), we can multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(4 - 2i\) is \(4 + 2i\).

Let's perform the calculation:

\[
\frac{6 + 3i}{4 - 2i} \times \frac{4 + 2i}{4 + 2i} = \frac{(6 + 3i)(4 + 2i)}{(4 - 2i)(4 + 2i)}
\]

### Step 1: Multiply the numerator
\[
(6 + 3i)(4 + 2i) = 6(4) + 6(2i) + 3i(4) + 3i(2i)
\]
\[
= 24 + 12i + 12i + 6i^2
\]
Since \(i^2 = -1\):
\[
= 24 + 24i + 6(-1) = 24 + 24i - 6 = 18 + 24i
\]

### Step 2: Multiply the denominator
\[
(4 - 2i)(4 + 2i) = 4^2 - (2i)^2 = 16 - 4i^2
\]
Since \(i^2 = -1\):
\[
= 16 - 4(-1) = 16 + 4 = 20
\]

### Step 3: Combine the results
Now, we have:
\[
\frac{18 + 24i}{20} = \frac{18}{20} + \frac{24i}{20} = 0.9 + 1.2i
\]

### Final answer:
\[
\frac{6 + 3i}{4 - 2i} = 0.9 + 1.2i
\]

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. Simplify \(\frac{5 + 2i}{3 - i}\).
2. What is the conjugate of \(7 + 4i\)?
3. Find the modulus of \(2 + 3i\).
4. Solve \((1 + i)^2\).
5. What is \(\frac{1 + 2i}{1 - 2i}\)?

**Tip:** Always multiply by the conjugate of the denominator when dividing complex numbers to eliminate the imaginary part in the denominator.

Learn how to simplify the complex division problem (6 + 3i) / (4 - 2i) by multiplying by the conjugate. This step-by-step solution covers key concepts in complex numbers and provides a detailed breakdown of the multiplication process. Suitable for Grades 9-12.

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