Question

divide.
\\(\frac{-3 + 5i}{5 + 2i}\\)
write your answer as a complex number in standard form.

Explanation:

Step1: Multiply numerator and denominator by conjugate

To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(5 + 2i\) is \(5 - 2i\). So we have:

$$ \frac{-3 + 5i}{5 + 2i} \times \frac{5 - 2i}{5 - 2i} $$

Step2: Expand numerator and denominator

First, expand the numerator using the distributive property (FOIL method):

$$
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$$

Since \(i^{2}=- 1\), we substitute that in:

$$ -15 + 31i-10\times(-1)=-15 + 31i + 10=-5 + 31i $$

Now expand the denominator using the formula \((a + b)(a - b)=a^{2}-b^{2}\), where \(a = 5\) and \(b = 2i\):

$$ (5 + 2i)(5 - 2i)=5^{2}-(2i)^{2}=25-4i^{2} $$

Substitute \(i^{2}=-1\):

$$ 25-4\times(-1)=25 + 4 = 29 $$

Step3: Write as complex number

Now we have \(\frac{-5 + 31i}{29}\), which we can split into the real and imaginary parts:

$$ \frac{-5}{29}+\frac{31}{29}i $$

Answer:

\(\frac{-5}{29}+\frac{31}{29}i\)