Question

question
express as a complex number in simplest a+bi form:
\\(\frac{-3 + 21i}{9 - 3i}\\)

Explanation:

Step1: Multiply numerator and denominator by conjugate

The conjugate of the denominator \(9 - 3i\) is \(9 + 3i\). Multiply the numerator and denominator by this conjugate:

$$ \frac{(-3 + 21i)(9 + 3i)}{(9 - 3i)(9 + 3i)} $$

Step2: Expand numerator and denominator

Expand the numerator using the distributive property (FOIL method):

$$
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$$

Expand the denominator using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\):

$$ 9^2-(3i)^2 = 81 - 9i^2 = 81 - 9(-1)=81 + 9 = 90 $$

Step3: Simplify the fraction

Now we have \(\frac{-90 + 180i}{90}\). Divide each term in the numerator by 90:

$$ \frac{-90}{90}+\frac{180i}{90}=-1 + 2i $$

Answer:

\(-1 + 2i\)