The problem in the image asks to find two complex numbers whose squares equal $5 - 12i$.

Let the complex number be of the form $z = a + bi$, where $a$ and $b$ are real numbers. To find $z$, we square it and equate it to $5 - 12i$.

Step-by-step approach:

1. $z^2 = (a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 - b^2 + 2abi$.
2. We know $z^2 = 5 - 12i$, so we equate real and imaginary parts:
   • Real part: $a^2 - b^2 = 5$
   • Imaginary part: $2ab = -12$

Now, let's solve this system of equations.

System of equations:

1. $a^2 - b^2 = 5$ (Real part)
2. $2ab = -12$ (Imaginary part)

From the second equation, solve for $b$ in terms of $a$: $b = \frac{-12}{2a} = \frac{-6}{a}$

Substituting into the first equation:

$a^2 - \left( \frac{-6}{a} \right)^2 = 5$ $a^2 - \frac{36}{a^2} = 5$

Now, let's solve this equation for $a$. I will calculate this next.The solutions for $a$ are $a = -3$, $a = 3$, $a = -2i$, and $a = 2i$.

However, since $a$ is a real number (for $z = a + bi$), we discard the imaginary solutions $a = -2i$ and $a = 2i$.

Thus, $a = 3$ or $a = -3$.

Now, solving for $b$:

Using $b = \frac{-6}{a}$:

• If $a = 3$, then $b = \frac{-6}{3} = -2$.
• If $a = -3$, then $b = \frac{-6}{-3} = 2$.

The two complex numbers are:

1. $z_1 = 3 - 2i$
2. $z_2 = -3 + 2i$

These are the two complex numbers whose squares equal $5 - 12i$.

Would you like more details or have any further questions?

Here are 5 related questions:

1. How do you verify that squaring $3 - 2i$ gives $5 - 12i$?
2. Can you explain why we equate real and imaginary parts separately?
3. How do we solve complex number equations geometrically?
4. What is the general form of a complex number and its conjugate?
5. What happens when you add and multiply two complex numbers?

Tip: Squaring a complex number involves expanding using the distributive property and remembering that $i^2 = -1$.