Problem 26 - Algebra and Trigonometry

Question

Divide and express the result in standard form. $$ \frac{-6 i}{3+2 i} $$

Step-by-Step Solution

Answer

The division of the given complex numbers results in \( \frac{12}{5} - \frac{18}{5}i \)

Step 1: Identify the Numerator and the Denominator

The numerator of the given complex fraction is \(-6 i\) and the denominator is \(3+2 i\). We aim to divide these two to get the result in standard form.

Step 2: Multiply by the Conjugate

To eliminate the imaginary part from the denominator, multiply both numerator and denominator by the conjugate of the denominator, which is \(3 - 2i\). The fraction then becomes \[ \frac{-6i(3 - 2i)}{(3 + 2i)(3 - 2i)} \]

Step 3: Simplify the Numerator

Multiplying the numbers in the numerator, we get: \[ -6i(3) + -6i(-2i) = -18i + 12 \], which simplifies to \(12 - 18i \)

Step 4: Simplify the Denominator

Multiplying the numbers in the denominator, we get: \[ (3)(3) + (3)(-2i) + (2i)(3) + (2i)(-2i) = 9 - 6i + 6i - 4 = 5 \]

Step 5: Divide by the Real Number

Now, simply divide the real and imaginary parts of the numerator by the real number in the denominator. The fraction becomes \( \frac{12}{5} - \frac{18i}{5}\)

Step 6: Express in Standard Form

The standard form for complex numbers is \(a + bi\), so our final answer is \( \frac{12}{5} - \frac{18}{5}i \)