To solve for the modulus of $z + \frac{1}{z}$, given that $z$ is a complex number satisfying $z - \frac{1}{z} = 4 + 3i$, we can follow these steps:

Step 1: Express $z$ in polar form

Let $z = r e^{i\theta}$, where $r$ is the modulus of $z$ and $\theta$ is the argument.

Step 2: Solve the equation $z - \frac{1}{z} = 4 + 3i$

First, express $\frac{1}{z}$: $\frac{1}{z} = \frac{1}{r e^{i\theta}} = \frac{1}{r} e^{-i\theta}$ Then, the given equation becomes: $r e^{i\theta} - \frac{1}{r} e^{-i\theta} = 4 + 3i$ This can be rewritten as: $r e^{i\theta} - \frac{1}{r} e^{-i\theta} = \left( r - \frac{1}{r} \right) \cos(\theta) + i \left( r + \frac{1}{r} \right) \sin(\theta) = 4 + 3i$ This implies that: $\left( r - \frac{1}{r} \right) \cos(\theta) = 4 \quad \text{and} \quad \left( r + \frac{1}{r} \right) \sin(\theta) = 3$

Step 3: Find $r$ and $\theta$

Square both sides of the equations and add them: $\left( \left( r - \frac{1}{r} \right) \cos(\theta) \right)^2 + \left( \left( r + \frac{1}{r} \right) \sin(\theta) \right)^2 = 4^2 + 3^2 = 25$ Expanding the squared terms: $\left( r - \frac{1}{r} \right)^2 \cos^2(\theta) + \left( r + \frac{1}{r} \right)^2 \sin^2(\theta) = 25$ This equation is quite complex to solve directly, but we proceed with a different method to simplify the calculations.

Step 4: Use properties of moduli

We can find the modulus of $z + \frac{1}{z}$ using the equation: $\left| z + \frac{1}{z} \right|^2 = \left( z + \frac{1}{z} \right) \left( \overline{z} + \frac{1}{\overline{z}} \right)$ Given $z - \frac{1}{z} = 4 + 3i$, the conjugate equation is $\overline{z} - \frac{1}{\overline{z}} = 4 - 3i$.

Now, multiply $(z + \frac{1}{z})$ by its conjugate: $\left| z + \frac{1}{z} \right|^2 = \left( z + \frac{1}{z} \right) \left( \overline{z} + \frac{1}{\overline{z}} \right) = |z|^2 + 2 + \frac{1}{|z|^2}$

We already know $|z|^2 - \frac{1}{|z|^2} = 4 + 3i$, so: $\text{Re}\left(z + \frac{1}{z}\right) = 4 \quad \text{and} \quad \text{Im}\left(z + \frac{1}{z}\right) = 3$

Finally, the modulus of $z + \frac{1}{z}$ is: $\left| z + \frac{1}{z} \right| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Thus, the closed-form solution for the modulus is 5.

If you have any questions, or want further details, feel free to ask!

Here are some related questions you might explore:

1. How do you compute the modulus of a complex number?
2. What is the geometric interpretation of $z + \frac{1}{z}$ for complex $z$?
3. How does the argument of $z$ affect $z - \frac{1}{z}$?
4. Can you find $z$ given $z - \frac{1}{z} = 4 + 3i$ explicitly?
5. How do you interpret $\left( z + \frac{1}{z} \right)$ in terms of the unit circle?

Tip: When dealing with complex numbers, it's often useful to work in polar form, as it simplifies multiplication and division.