Question

express as a complex number in simplest a+bi form: \\(\frac{-9 - 3i}{-3 + 2i}\\)

Explanation:

Step1: Multiply numerator and denominator by the conjugate of the denominator

The conjugate of \(-3 + 2i\) is \(-3 - 2i\). So we have:
\[
\frac{(-9 - 3i)(-3 - 2i)}{(-3 + 2i)(-3 - 2i)}
\]

Step2: Expand the numerator

Using the distributive property (FOIL method):
\[

$$\begin{align*} (-9 - 3i)(-3 - 2i)&= (-9)(-3)+(-9)(-2i)+(-3i)(-3)+(-3i)(-2i)\\ &= 27 + 18i + 9i + 6i^2\\ &= 27 + 27i + 6(-1) \quad (\text{since } i^2 = -1)\\ &= 27 + 27i - 6\\ &= 21 + 27i \end{align*}$$

\]

Step3: Expand the denominator

Using the difference of squares formula \((a + b)(a - b)=a^2 - b^2\):
\[

$$\begin{align*} (-3 + 2i)(-3 - 2i)&= (-3)^2-(2i)^2\\ &= 9 - 4i^2\\ &= 9 - 4(-1)\\ &= 9 + 4\\ &= 13 \end{align*}$$

\]

Step4: Simplify the fraction

Now we have \(\frac{21 + 27i}{13}\), which can be written as:
\[
\frac{21}{13}+\frac{27}{13}i
\]

Answer:

\(\frac{21}{13}+\frac{27}{13}i\)