`(3i +2)/(-i-3)`
If the expression above is expressed in the form `a +bi,` where `I = sqrt-1,` what is the value of b ?

Options

A. `-0.7`
B. `0.7`
C. `-0.9`
D. `0.9`

Answer

To solve the expression \((3i + 2)/(-i - 3)\) and express it in the form \(a + bi\), we will follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ \frac{3i + 2}{-i - 3} \]

### Step 2: Multiply by the conjugate
To simplify the expression, we multiply the numerator and denominator by the conjugate of the denominator, which is \(-i + 3\):
\[ \frac{(3i + 2)(-i + 3)}{(-i - 3)(-i + 3)} \]

### Step 3: Simplify the denominator
Using the difference of squares, we simplify the denominator:
\[ (-i - 3)(-i + 3) = (-i)^2 - 3^2 = -1 - 9 = -10 \]

### Step 4: Expand the numerator
Now we expand the numerator:
\[ (3i + 2)(-i + 3) = 3i \cdot (-i) + 3 \cdot 3 + 2 \cdot (-i) + 2 \cdot 3 \]
Calculating each term:
- \(3i \cdot (-i) = -3i^2 = 3\) (since \(i^2 = -1\))
- \(3 \cdot 3 = 9\)
- \(2 \cdot (-i) = -2i\)
- \(2 \cdot 3 = 6\)

Combining these:
\[ 3 + 9 - 2i + 6 = 18 - 2i \]

### Step 5: Combine the results
Now we substitute back into our expression:
\[ \frac{18 - 2i}{-10} \]
This simplifies to:
\[ \frac{18}{-10} - \frac{2i}{-10} = -\frac{9}{5} + \frac{1}{5}i \]

### Step 6: Identify \(a\) and \(b\)
Now we can express this in the form \(a + bi\):
\[ -\frac{9}{5} + \frac{1}{5}i \]
Here, \(a = -\frac{9}{5}\) and \(b = \frac{1}{5}\).

### Final Answer
Thus, the value of \(b\) is:
\[ \frac{1}{5} = 0.2 \]