To square the complex number \( (2 - 3i)^2 \), we can use the formula for squaring a binomial, \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = 2\) and \(b = -3i\).

Let's go through the steps:

1. **Square the first term**: \(2^2 = 4\)
2. **Multiply the first and second term by 2**: \(2 \times 2 \times (-3i) = -12i\)
3. **Square the second term**: \((-3i)^2 = 9i^2\)

Now, combine these results:

\[ 4 - 12i + 9i^2 \]

Remembering that \(i^2 = -1\):

\[ 4 - 12i + 9(-1) \]

\[ 4 - 12i - 9 \]

Combine the real parts:

\[ 4 - 9 = -5 \]

So, we get:

\[ -5 - 12i \]

Therefore, \((2 - 3i)^2 = -5 - 12i\).

Would you like more details or have any questions?

Here are 8 related questions you might find interesting:
1. How do you expand \((a + bi)^2\) in general?
2. What is the geometric interpretation of squaring a complex number?
3. How do you simplify higher powers of complex numbers?
4. What is the significance of the imaginary unit \(i\)?
5. How do you add and subtract complex numbers?
6. What is the polar form of a complex number?
7. How do you multiply complex numbers using their polar forms?
8. How do you find the inverse of a complex number?

**Tip:** When dealing with powers of complex numbers, remember to use the properties of \(i\), such as \(i^2 = -1\), to simplify your calculations.

Explore how to compute the square of the complex number (2 - 3i) with a detailed step-by-step solution. This problem involves complex number arithmetic and the binomial square formula. Suitable for students in Grades 10-12.

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