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The given expression is \((12 - 9i)^2\), and you need to simplify it into the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.

To simplify:

1. Use the formula \((x - y)^2 = x^2 - 2xy + y^2\).
   - Here, \(x = 12\) and \(y = 9i\).
   
2. Expand \((12 - 9i)^2\):

\[
(12 - 9i)^2 = 12^2 - 2(12)(9i) + (9i)^2
\]

Now, simplify each term:
- \(12^2 = 144\),
- \(-2(12)(9i) = -216i\),
- \((9i)^2 = 81i^2 = 81(-1) = -81\) (since \(i^2 = -1\)).

3. Combine these results:

\[
(12 - 9i)^2 = 144 - 216i - 81
\]

4. Simplify further:

\[
144 - 81 = 63
\]

So the simplified expression is:

\[
63 - 216i
\]

Thus, the expression in \(a + bi\) form is:

\[
63 - 216i
\]

Would you like further explanation or details on this process?

Here are 5 related questions to expand your understanding:
1. How do you square a complex number using binomial expansion?
2. What is the significance of the imaginary unit \(i\) in complex numbers?
3. How does \(i^2 = -1\) affect calculations involving complex numbers?
4. Can you express the product of two complex numbers geometrically?
5. What are the real and imaginary components of a complex number?

**Tip:** When squaring complex numbers, always remember that \(i^2 = -1\), which simplifies the calculation.

Simplify the expression to a + bi form: (12 - 9i)^2

Learn how to simplify the complex number expression (12 - 9i)^2 into a + bi form with a detailed step-by-step solution. This problem covers complex numbers, binomial expansion, and properties of the imaginary unit i.

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