The equation in the image appears to be:

$Z + i|Z| = 12 + 9i$

where $Z$ is a complex number, $i$ is the imaginary unit, and $|Z|$ is the modulus (magnitude) of $Z$.

Let $Z = x + yi$, where $x$ is the real part of $Z$ and $y$ is the imaginary part of $Z$.

We can break this problem into real and imaginary parts:

Step 1: Substitute $Z = x + yi$ into the equation

The modulus $|Z|$ is given by:

$|Z| = \sqrt{x^2 + y^2}$

Substituting $Z = x + yi$ into the original equation:

$(x + yi) + i\sqrt{x^2 + y^2} = 12 + 9i$

Step 2: Separate into real and imaginary parts

Now, equate the real and imaginary parts of both sides of the equation:

• Real part: $x = 12$

• Imaginary part: $y + \sqrt{x^2 + y^2} = 9$

Step 3: Solve for $y$

From the real part, we know $x = 12$. Substitute $x = 12$ into the equation for the imaginary part:

$y + \sqrt{12^2 + y^2} = 9$ $y + \sqrt{144 + y^2} = 9$

Now solve for $y$:

$\sqrt{144 + y^2} = 9 - y$ Square both sides: $144 + y^2 = (9 - y)^2$ Expand the right-hand side: $144 + y^2 = 81 - 18y + y^2$ Simplify the equation: $144 = 81 - 18y$ $144 - 81 = -18y$ $63 = -18y$ $y = -\frac{63}{18} = -\frac{7}{2} = -3.5$

Step 4: Final answer

The imaginary part of $Z$ is $y = -3.5$.

Would you like more details or have any questions? Here are some related questions:

1. What is the modulus of the complex number $Z$?
2. How do you calculate the argument of a complex number?
3. Can we verify the solution by substituting back into the original equation?
4. How is the modulus $|Z|$ affected if both the real and imaginary parts change?
5. What geometric interpretation can be given for complex numbers in this equation?

Tip: When solving complex equations, always separate real and imaginary parts for clearer analysis.