Question

divide.
\\(\frac{-6i}{-3 + 4i}\\)
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 \(-3 + 4i\) is \(-3 - 4i\). So we have:

$$ \frac{-6i}{-3 + 4i} \times \frac{-3 - 4i}{-3 - 4i} $$

Step2: Multiply the numerators.

Using the distributive property (FOIL for complex numbers), \((-6i)(-3 - 4i) = (-6i)(-3) + (-6i)(-4i) = 18i + 24i^2\). Since \(i^2 = -1\), this becomes \(18i + 24(-1) = -24 + 18i\).

Step3: Multiply the denominators.

Using the formula \((a + b)(a - b) = a^2 - b^2\), where \(a = -3\) and \(b = 4i\), we get \((-3)^2 - (4i)^2 = 9 - 16i^2\). Substituting \(i^2 = -1\), we have \(9 - 16(-1) = 9 + 16 = 25\).

Step4: Write the result as a complex number.

Now we have \(\frac{-24 + 18i}{25}\), which can be split into \(\frac{-24}{25} + \frac{18}{25}i\).

Answer:

\(\frac{-24}{25} + \frac{18}{25}i\)