Questions: Divide the following complex numbers. -2+5i / -3+4i Select the correct answer below: -14-7i / -4 -14+7i / -4 26-7i / 25 26+7i / 25

Transcript text: Divide the following complex numbers. \[ \frac{-2+5 i}{-3+4 i} \] Select the correct answer below: $\frac{-14-7 i}{-4}$ $\frac{-14+7 i}{-4}$ $\frac{26-7 i}{25}$ $\frac{26+7 i}{25}$

Solution

Solution Steps

Step 1: Define the Complex Numbers

Let $ z_1 = -2 + 5i $ and $ z_2 = -3 + 4i $.

Step 2: Multiply by the Conjugate

To divide $ \frac{z_1}{z_2} $, multiply both the numerator and the denominator by the conjugate of the denominator $ z_2^* = -3 - 4i $: \[ \frac{z_1}{z_2} = \frac{(-2 + 5i)(-3 - 4i)}{(-3 + 4i)(-3 - 4i)} \]

Step 3: Calculate the Numerator

Calculating the numerator: \[ (-2 + 5i)(-3 - 4i) = (-2)(-3) + (-2)(-4i) + (5i)(-3) + (5i)(-4i) = 6 + 8i - 15i - 20 = 26 - 7i \]

Step 4: Calculate the Denominator

Calculating the denominator: \[ (-3 + 4i)(-3 - 4i) = (-3)^2 + (4i)^2 = 9 + 16 = 25 \]

Step 5: Combine the Results

Now, we can express the division as: \[ \frac{26 - 7i}{25} \]

Step 6: Simplify the Expression

This can be simplified to: \[ \frac{26}{25} - \frac{7}{25}i \]

Thus, the result of the division is $ \frac{26}{25} - \frac{7}{25}i $.