The solutions to the given equation are:

$z = -\frac{1}{2} + \frac{5i}{2} \quad \text{and} \quad z = \frac{4}{17} + \frac{35i}{17}$

Would you like further details or explanations? Here are some related questions to consider:

1. How can this solution be verified step-by-step?
2. What is the significance of the imaginary unit $i$ in complex equations?
3. How can this method be applied to other complex equations?
4. Can we graph these solutions on the complex plane?
5. What are some real-world applications of solving complex number equations?

Tip: When solving complex equations, always check for conjugate pairs as potential solutions.