To evaluate the quotient $\frac{20i}{1 - 3i}$ and express it in the form $a + bi$, follow these steps:

Step 1: Multiply by the conjugate

To eliminate the imaginary part in the denominator, multiply both the numerator and denominator by the conjugate of $1 - 3i$, which is $1 + 3i$. The conjugate reverses the sign of the imaginary part.

$\frac{20i}{1 - 3i} \times \frac{1 + 3i}{1 + 3i} = \frac{20i(1 + 3i)}{(1 - 3i)(1 + 3i)}$

Step 2: Simplify the denominator

The denominator is of the form $(a - b)(a + b)$, which simplifies as $a^2 - b^2$, where $a = 1$ and $b = 3i$. This results in:

$(1 - 3i)(1 + 3i) = 1^2 - (3i)^2 = 1 - 9(-1) = 1 + 9 = 10$

So, the denominator is 10.

Step 3: Expand the numerator

Now, expand the numerator:

$20i(1 + 3i) = 20i + 60i^2$

Since $i^2 = -1$, substitute $-1$ for $i^2$:

$20i + 60(-1) = 20i - 60$

So, the numerator becomes $-60 + 20i$.

Step 4: Simplify the expression

Now divide each term of the numerator by 10:

$\frac{-60 + 20i}{10} = \frac{-60}{10} + \frac{20i}{10} = -6 + 2i$

Final Answer:

The result in the form $a + bi$ is:

$\boxed{-6 + 2i}$

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Would you like more details on any of the steps? Here are 5 related questions to help deepen your understanding:

1. How do you multiply complex numbers in polar form?
2. What is the geometric interpretation of multiplying a complex number by its conjugate?
3. How does multiplying by the conjugate simplify division of complex numbers?
4. What happens when you add or subtract two complex conjugates?
5. How can we represent complex numbers on the complex plane?

Tip: Always use the conjugate to simplify division involving complex numbers!