Let's solve the equation \(\frac{1+2i}{3-4i} + \frac{2-i}{5i} = -\frac{2}{5}\).

### Step 1: Simplify \(\frac{1+2i}{3-4i}\)

To simplify \(\frac{1+2i}{3-4i}\), multiply the numerator and the denominator by the conjugate of the denominator, which is \(3 + 4i\):

\[
\frac{1+2i}{3-4i} \times \frac{3+4i}{3+4i} = \frac{(1+2i)(3+4i)}{(3-4i)(3+4i)}
\]

First, compute the denominator:

\[
(3-4i)(3+4i) = 3^2 - (4i)^2 = 9 - (-16) = 9 + 16 = 25
\]

Now, compute the numerator:

\[
(1+2i)(3+4i) = 1(3) + 1(4i) + 2i(3) + 2i(4i) = 3 + 4i + 6i + 8i^2
\]

Since \(i^2 = -1\):

\[
3 + 4i + 6i - 8 = -5 + 10i
\]

Thus:

\[
\frac{1+2i}{3-4i} = \frac{-5+10i}{25} = -\frac{1}{5} + \frac{2i}{5}
\]

### Step 2: Simplify \(\frac{2-i}{5i}\)

To simplify \(\frac{2-i}{5i}\), multiply the numerator and the denominator by \(i\) (since \(i\) is the conjugate of \(i\)):

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

Compute the numerator:

\[
(2-i)i = 2i - i^2 = 2i + 1 \quad \text{(since \(i^2 = -1\))}
\]

The denominator becomes:

\[
5i^2 = 5(-1) = -5
\]

Thus:

\[
\frac{2i + 1}{-5} = -\frac{1}{5} - \frac{2i}{5}
\]

### Step 3: Combine the simplified fractions

Now, add the two simplified expressions:

\[
\left(-\frac{1}{5} + \frac{2i}{5}\right) + \left(-\frac{1}{5} - \frac{2i}{5}\right)
\]

Combine like terms:

\[
-\frac{1}{5} - \frac{1}{5} + \frac{2i}{5} - \frac{2i}{5} = -\frac{2}{5}
\]

### Step 4: Conclusion

The left-hand side simplifies to \(-\frac{2}{5}\), which is equal to the right-hand side of the equation. Therefore, the equation holds true:

\[
\frac{1+2i}{3-4i} + \frac{2-i}{5i} = -\frac{2}{5}
\]

Would you like more details on any part of this process or have any questions?

---

Here are some related questions to consider:

1. How do you find the conjugate of a complex number?
2. What is the process for rationalizing the denominator in complex fractions?
3. How do you add and subtract complex numbers?
4. What are some common applications of complex numbers in physics and engineering?
5. How can you visualize complex numbers on the complex plane?

**Tip:** When working with complex fractions, always multiply by the conjugate to simplify and eliminate the imaginary unit from the denominator.

Learn how to solve the complex fraction equation (1+2i)/(3-4i) + (2-i)/(5i) = -2/5 step-by-step. Understand complex numbers, their conjugates, and simplifying complex fractions. Suitable for advanced high school or college-level students.

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