Please help me by showing alternate methods to solve this complex number SAT question
I encountered this question in one of my SAT practice tests.
I know the answer is option C, however the only way I got the answer was by trial and error of trying multiple ways to simplify the equation and I ended up rationalising it to get answer choice C.
Is there any other, perhaps easier or more direct method, that I can use to solve these types of questions?
Thank you in advance 🙂
Best Answer
Two ways to approach this problem. First: As quasi suggests in the comments, multiply by the conjugate of the denominator.
\begin{align} \frac{3-5i}{8+2i} & = \frac{3-5i}{8+2i} \times \frac{8-2i}{8-2i} \\ & = \frac{24-40i-6i-10}{8^2+2^2} \\ & = \frac{14-46i}{68} = \frac{7-23i}{34} \end{align}
The second approach, given that it's a multiple choice problem, is to multiply each of the answers by $8+2i$ and see if you obtain $3-5i$. I think the first approach is simpler, but they'll both work.