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We are given the complex numbers \( z_1 = 1 + 3i \) and \( z_2 = 2 - i \), and we are asked to express \( \frac{z_1 + z_2}{z_1 z_2} \) in the form \( a + bi \), where \( a \) and \( b \) are real numbers.

### Step 1: Calculate \( z_1 + z_2 \)

\[
z_1 + z_2 = (1 + 3i) + (2 - i) = 3 + 2i
\]

### Step 2: Calculate \( z_1 z_2 \)

We use the distributive property to multiply \( z_1 \) and \( z_2 \):

\[
z_1 z_2 = (1 + 3i)(2 - i)
\]
\[
= 1 \cdot 2 + 1 \cdot (-i) + 3i \cdot 2 + 3i \cdot (-i)
\]
\[
= 2 - i + 6i - 3i^2
\]

Since \( i^2 = -1 \), this becomes:

\[
2 - i + 6i + 3 = 5 + 5i
\]

### Step 3: Divide \( z_1 + z_2 \) by \( z_1 z_2 \)

We now compute \( \frac{3 + 2i}{5 + 5i} \). To do this, multiply the numerator and denominator by the conjugate of the denominator:

\[
\frac{3 + 2i}{5 + 5i} \times \frac{5 - 5i}{5 - 5i} = \frac{(3 + 2i)(5 - 5i)}{(5 + 5i)(5 - 5i)}
\]

First, calculate the denominator:

\[
(5 + 5i)(5 - 5i) = 25 - (5i)^2 = 25 - (-25) = 50
\]

Now, calculate the numerator:

\[
(3 + 2i)(5 - 5i) = 3 \cdot 5 + 3 \cdot (-5i) + 2i \cdot 5 + 2i \cdot (-5i)
\]
\[
= 15 - 15i + 10i - 10i^2
\]
\[
= 15 - 15i + 10i + 10 = 25 - 5i
\]

Thus, we have:

\[
\frac{25 - 5i}{50} = \frac{25}{50} - \frac{5i}{50} = \frac{1}{2} - \frac{1}{10}i
\]

### Final Answer:
The expression in the form \( a + bi \) is \( \frac{1}{2} - \frac{1}{10}i \). This matches option (c).

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Would you like further details on this solution?

Here are five related questions:
1. How do you multiply two complex numbers?
2. What is the geometric interpretation of multiplying complex numbers?
3. How do you find the conjugate of a complex number and why is it useful?
4. How do you divide complex numbers in polar form?
5. How can you express a complex number in polar and rectangular forms?

**Tip:** Always simplify complex expressions step by step, especially when multiplying or dividing by a conjugate!

Given z₁ = 1 + 3i and z₂ = 2 - i, express (z₁ + z₂) / (z₁ z₂) in the form of a + bi, where a and b are real numbers.

Learn how to divide complex numbers using conjugates by solving (z₁ + z₂) / (z₁ z₂) where z₁ = 1 + 3i and z₂ = 2 - i. This problem involves complex number addition, multiplication, and division, with a step-by-step breakdown. Suitable for Grades 10-12.

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