Simplify to a + bi: -2/(3 - i) * (5/i) =

Steps to Solve

1. Multiply by the Conjugate of the Denominator

We first tackle the fraction $-\frac{2}{3 - i}$. To eliminate the complex denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is $3 + i$:

$$ -\frac{2}{3 - i} \cdot \frac{3 + i}{3 + i} = -\frac{2(3 + i)}{(3 - i)(3 + i)} $$

2. Simplify the Denominator

Next, we simplify the denominator using the difference of squares:

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

Thus, our expression now becomes:

$$ -\frac{2(3 + i)}{10} $$

3. Simplify the Expression

Now we can simplify the fraction:

$$ -\frac{2(3 + i)}{10} = -\frac{6 + 2i}{10} = -\frac{3}{5} - \frac{1}{5} i $$

4. Multiply the Whole Expression by $5/i$

Next, we multiply our result by $\frac{5}{i}$:

$$ \left(-\frac{3}{5} - \frac{1}{5} i\right) \cdot \frac{5}{i} $$

5. Distribute the Terms

Distributing yields:

$$ -\frac{3}{5} \cdot \frac{5}{i} - \frac{1}{5} i \cdot \frac{5}{i} $$

This simplifies to:

$$ -\frac{3}{i} - 1 $$

6. Rewrite $\frac{3}{i}$

To express $-\frac{3}{i}$ in standard form, multiply by $\frac{i}{i}$:

$$ -\frac{3}{i} = -3 \cdot \frac{i}{i^2} = -3 \cdot \frac{i}{-1} = 3i $$

7. Combine the Real and Imaginary Parts

Now we combine the simplified terms:

$$ 3i - 1 = -1 + 3i $$

Thus, we have the final result in the form $a + bi$.