Questions: Divide the following two complex numbers. (4-3 i)/(3+2 i) a.) (6/5)-(17/5) i b.) (18/5)-(17/5) i c.) (6/13)-(17/13) i d.) (18/13)-(17/13) i

Divide the following two complex numbers. (4-3 i)/(3+2 i) a.) (6/5)-(17/5) i b.) (18/5)-(17/5) i c.) (6/13)-(17/13) i d.) (18/13)-(17/13) i
Transcript text: Divide the following two complex numbers. \[ \frac{4-3 i}{3+2 i} \] a.) $\frac{6}{5}-\frac{17}{5} i$ b.) $\frac{18}{5}-\frac{17}{5} i$ c.) $\frac{6}{13}-\frac{17}{13} i$ d.) $\frac{18}{13}-\frac{17}{13} i$
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Solution

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Solution Steps

Step 1: Define the Complex Numbers

Let z1=4−3i z_1 = 4 - 3i (the numerator) and z2=3+2i z_2 = 3 + 2i (the denominator).

Step 2: Find the Conjugate of the Denominator

The conjugate of the denominator z2 z_2 is z2‾=3−2i \overline{z_2} = 3 - 2i .

Step 3: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator: z1z2=(4−3i)(3−2i)(3+2i)(3−2i) \frac{z_1}{z_2} = \frac{(4 - 3i)(3 - 2i)}{(3 + 2i)(3 - 2i)}

Step 4: Simplify the Denominator

Calculate the denominator: (3+2i)(3−2i)=32−(2i)2=9−(−4)=9+4=13 (3 + 2i)(3 - 2i) = 3^2 - (2i)^2 = 9 - (-4) = 9 + 4 = 13

Step 5: Expand the Numerator

Expand the numerator: (4−3i)(3−2i)=4⋅3+4⋅(−2i)−3i⋅3−3i⋅(−2i)=12−8i−9i+6=18−17i (4 - 3i)(3 - 2i) = 4 \cdot 3 + 4 \cdot (-2i) - 3i \cdot 3 - 3i \cdot (-2i) = 12 - 8i - 9i + 6 = 18 - 17i

Step 6: Combine the Results

Now, combine the results: 18−17i13=1813−1713i \frac{18 - 17i}{13} = \frac{18}{13} - \frac{17}{13}i

Step 7: Final Result

Thus, the result of dividing the two complex numbers is: 1813−1713i \frac{18}{13} - \frac{17}{13}i

Final Answer

The correct answer is D: 1813−1713i \boxed{\frac{18}{13} - \frac{17}{13} i}

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