To simplify the given expression $(12 - 9i)^2$ into $a + bi$ form, we proceed as follows:

Step 1: Expand the square using the binomial theorem

$(12 - 9i)^2 = (12)^2 - 2(12)(9i) + (9i)^2$

Step 2: Simplify each term

1. $(12)^2 = 144$
2. $-2(12)(9i) = -216i$
3. $(9i)^2 = (9)^2(i^2) = 81(-1) = -81$

Step 3: Combine like terms

Now combine the real and imaginary parts: $(12 - 9i)^2 = 144 - 81 - 216i$

Simplify the real part: $144 - 81 = 63$

Thus, the expression becomes: $63 - 216i$

Final Answer:

$(12 - 9i)^2 = 63 - 216i$

Verification:

This is in the standard $a + bi$ form, where:

• $a = 63$
• $b = -216$

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Would you like to see a detailed explanation of a specific step or have additional questions?
Here are 5 related questions to explore further:

1. What is the significance of $i^2 = -1$ in simplifying complex numbers?
2. How do you find the magnitude of the complex number $12 - 9i$?
3. What happens if we compute $(12 + 9i)^2$ instead of $(12 - 9i)^2$?
4. How can we express the result in polar form instead of $a + bi$?
5. What are some common errors to avoid when expanding $(a + bi)^2$?

Tip: When squaring a complex number, carefully handle the $i^2$ term—it always simplifies to $-1$.