Question

divide.
\frac{6 - 5i}{-5 - 3i}
write your answer as a complex number in standard form.

Explanation:

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

The conjugate of \(-5 - 3i\) is \(-5 + 3i\). So we have:

$$ \frac{(6 - 5i)(-5 + 3i)}{(-5 - 3i)(-5 + 3i)} $$

Step2: Expand the numerator and the denominator.

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

$$
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$$

Since \(i^2=-1\), we substitute:

$$ -30 + 43i - 15\times(-1)=-30 + 43i + 15=-15 + 43i $$

Now, expand the denominator using the difference of squares formula \((a + b)(a - b)=a^2 - b^2\), where \(a=-5\) and \(b = 3i\):

$$ (-5 - 3i)(-5 + 3i)=(-5)^2-(3i)^2=25 - 9i^2 $$

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

$$ 25 - 9\times(-1)=25 + 9 = 34 $$

Step3: Write the fraction as a complex number.

Now we have \(\frac{-15 + 43i}{34}\), which can be written as \(-\frac{15}{34}+\frac{43}{34}i\)

Answer:

\(-\frac{15}{34}+\frac{43}{34}i\)