Question: Divide the following two complex numbers.
$3 + 2i$
───────
$5 - 3i$
a.) $\frac{9}{16} + \frac{19}{16} i$
b.) $\frac{9}{34} + \frac{19}{34} i$
c.) $\frac{21}{34} + \frac{19}{34} i$
d.) $\frac{21}{16} + \frac{19}{16} i$
1. **State the problem:** We need to divide the complex number $3 + 2i$ by $5 - 3i$.
2. **Formula and rule:** To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.
The conjugate of $5 - 3i$ is $5 + 3i$.
3. **Multiply numerator and denominator:**
$$\frac{3 + 2i}{5 - 3i} \times \frac{5 + 3i}{5 + 3i} = \frac{(3 + 2i)(5 + 3i)}{(5 - 3i)(5 + 3i)}$$
4. **Calculate numerator:**
$$(3)(5) + (3)(3i) + (2i)(5) + (2i)(3i) = 15 + 9i + 10i + 6i^2$$
Since $i^2 = -1$, this becomes:
$$15 + 9i + 10i + 6(-1) = 15 + 19i - 6 = 9 + 19i$$
5. **Calculate denominator:**
$$(5)^2 - (3i)^2 = 25 - 9i^2 = 25 - 9(-1) = 25 + 9 = 34$$
6. **Write the fraction:**
$$\frac{9 + 19i}{34} = \frac{9}{34} + \frac{19}{34}i$$
7. **Final answer:**
The quotient is $\boxed{\frac{9}{34} + \frac{19}{34}i}$, which corresponds to option b.)