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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 \( i^2 = -1 \).

Find two complex numbers whose squares equal 5 - 12i.

Learn how to find two complex numbers whose squares equal 5 - 12i with step-by-step solutions. This problem covers key concepts in complex numbers, equating real and imaginary parts, and solving systems of equations. Suitable for advanced high school or early college students.

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