Question

what is the quotient of \\(\frac{22 + 3i}{5 + 2i}\\)? the quotient is \\(\boxed{}\\).

Explanation:

Step1: Identify the conjugate denominator

The conjugate of $5+2i$ is $5-2i$.

Step2: Multiply numerator and denominator by conjugate

$$\frac{22+3i}{5+2i} \times \frac{5-2i}{5-2i} = \frac{(22+3i)(5-2i)}{(5+2i)(5-2i)}$$

Step3: Expand numerator using FOIL

$$(22)(5) + (22)(-2i) + (3i)(5) + (3i)(-2i) = 110 -44i +15i -6i^2$$
Since $i^2=-1$, substitute:
$$110 -29i -6(-1) = 110 -29i +6 = 116-29i$$

Step4: Expand denominator (difference of squares)

$$5^2 - (2i)^2 = 25 -4i^2 = 25 -4(-1) = 25+4=29$$

Step5: Simplify the fraction

$$\frac{116-29i}{29} = \frac{116}{29} - \frac{29i}{29} = 4 - i$$

Answer:

$4 - i$