To divide the complex numbers [tex]\(\frac{4 - 4i}{-2 - 3i}\)[/tex], we'll follow these steps and provide the result in the form [tex]\(a + bi\)[/tex]:
1. Conjugate of the Denominator: Multiply both the numerator and the denominator by the conjugate of the denominator to remove the imaginary part in the denominator. The conjugate of [tex]\(-2 - 3i\)[/tex] is [tex]\(-2 + 3i\)[/tex].
2. Numerator Multiplication:
[tex]\[ (4 - 4i)(-2 + 3i) \][/tex]
3. Denominator Multiplication:
[tex]\[
(-2 - 3i)(-2 + 3i)
\][/tex]
4. Perform the Multiplications:
- For the numerator:
[tex]\[
(4 - 4i)(-2 + 3i) = 4(-2) + 4(3i) - 4i(-2) - 4i(3i)
\][/tex]
Simplifying this:
[tex]\[
= -8 + 12i + 8i - 12i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= -8 + 20i + 12
\][/tex]
Combining real and imaginary parts:
[tex]\[
= 4 + 20i
\][/tex]
- For the denominator:
[tex]\[
(-2 - 3i)(-2 + 3i) = (-2)^2 - (3i)^2
\][/tex]
Simplifying this:
[tex]\[
= 4 - 9(-1)
\][/tex]
Since [tex]\(i^2 = -1\)[/tex]:
[tex]\[
= 4 + 9 = 13
\][/tex]
5. Division:
Now we divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{4 + 20i}{13}
\][/tex]
Separating real and imaginary parts:
[tex]\[
\frac{4}{13} + \frac{20i}{13}
\][/tex]
Thus, in the form [tex]\(a + bi\)[/tex], the result is:
[tex]\[
a = \frac{4}{13}, \quad b = \frac{20}{13}
\][/tex]
To present the final answer more clearly:
[tex]\[
a \approx 0.30769230769230776
\][/tex]
[tex]\[
b \approx 1.5384615384615383
\][/tex]
Hence:
[tex]\[
\frac{4-4i}{-2-3i} = 0.30769230769230776 + 1.5384615384615383i
\][/tex]
[tex]\[
\begin{array}{l}
a=0.30769230769230776 \\
b=1.5384615384615383
\end{array}
\][/tex]
These values match what we found from the division process.
