To express $(1 - \sqrt{3}i)^2$ in the form $a + bi$, where $a, b \in \mathbb{R}$, we need to expand and simplify the given expression.

Step 1: Expand the expression

We start by expanding $(1 - \sqrt{3}i)^2$ using the binomial theorem:

$(1 - \sqrt{3}i)^2 = 1^2 - 2(1)(\sqrt{3}i) + (\sqrt{3}i)^2$

Step 2: Simplify each term

1. $1^2 = 1$
2. $-2(1)(\sqrt{3}i) = -2\sqrt{3}i$
3. $(\sqrt{3}i)^2 = (\sqrt{3})^2 \times i^2 = 3 \times (-1) = -3$

Step 3: Combine the results

Now, combining all these terms:

$(1 - \sqrt{3}i)^2 = 1 - 2\sqrt{3}i - 3$

This simplifies to:

$-2 - 2\sqrt{3}i$

So, $(1 - \sqrt{3}i)^2 = -2 - 2\sqrt{3}i$.

Final Answer

The expression $(1 - \sqrt{3}i)^2$ in the form $a + bi$ is:

$\boxed{-2 - 2\sqrt{3}i}$

Do you want details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the result change if the original expression was $(1 + \sqrt{3}i)^2$?
2. What is the magnitude of the complex number $-2 - 2\sqrt{3}i$?
3. How can we express $(1 - \sqrt{3}i)^3$ in the form $a + bi$?
4. What is the polar form of the complex number $1 - \sqrt{3}i$?
5. How does the angle (argument) of $1 - \sqrt{3}i$ relate to the angle of $-2 - 2\sqrt{3}i$?

Tip: When dealing with complex numbers, always remember to simplify the imaginary unit $i^2 = -1$ as it often changes the sign of terms.