The imaginary number "I" is such that `i^(2)=-1`. Which of the following options is equivalent to `((1-2i)/(2+3i))` ?

Options

A. `-(4)/(13)-i/13`
B. `(4)/(13)+i/13`
C. 8-7i
D. `(8)/(13)+i`

Answer

To solve the expression \(\frac{1 - 2i}{2 + 3i}\), we will multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + 3i\) is \(2 - 3i\).

### Step 1: Multiply by the conjugate
\[ \frac{1 - 2i}{2 + 3i} \cdot \frac{2 - 3i}{2 - 3i} \]

### Step 2: Expand the numerator
Using the distributive property (FOIL method):
\[ (1 - 2i)(2 - 3i) = 1 \cdot 2 + 1 \cdot (-3i) - 2i \cdot 2 - 2i \cdot (-3i) \]
Calculating each term:
- \(1 \cdot 2 = 2\)
- \(1 \cdot (-3i) = -3i\)
- \(-2i \cdot 2 = -4i\)
- \(-2i \cdot (-3i) = 6i^2\) (and since \(i^2 = -1\), this becomes \(6(-1) = -6\))

Now, combine these results:
\[ 2 - 3i - 4i - 6 = 2 - 6 - 7i = -4 - 7i \]

### Step 3: Expand the denominator
Using the formula \(a^2 - b^2\):
\[ (2 + 3i)(2 - 3i) = 2^2 - (3i)^2 = 4 - 9i^2 \]
Since \(i^2 = -1\), we have:
\[ 4 - 9(-1) = 4 + 9 = 13 \]

### Step 4: Combine the results
Now we can write the expression as:
\[ \frac{-4 - 7i}{13} \]

### Step 5: Split the fraction
This can be separated into real and imaginary parts:
\[ -\frac{4}{13} - \frac{7}{13}i \]

Thus, the final result is:
\[ -\frac{4}{13} - \frac{7}{13}i \]