In this the answer is simple: .
Square roots occur as negatives,
As part of a question I have to find the square root of -24 + 10i. I know that 1 + 5i is one square root, but I have no idea how to find this other than guess and check. I tried converting to polar form, but yet again it's the argument that's getting me. I know that x = -24, y = 10, and r = 26. But there doesn't seem to be a "nice" argument based on those values. arctan(10/-24) = 157.380... degrees, arccos(-24/26) = 157.380... degrees, and arcsin(10/26) = 157.380... degrees. So it's consistent, but it seems like there's no way it's going to give me an exact value for the square root, even though I know that there's a 'nice' root (1 + 5i). I thought about maybe just leaving the argument as (arccos(-24/26)) and then manipulating double angle identities, but I wasn't sure how to properly do this. Any help?
I guess my question wasn't clear enough. I understand that (-1-5i) would also be a square root, I just don't understand how to find +/-(1+5i) in the first place other than guess and check. The method I was taught was to convert everything into polar form, but (-24+10i) doesn't seem to convert nicely, and decimal answers won't do, I need exact answers.
I've substituted these values back into equation (2) to get a value for a, and then used that value to form the complex number which is one of the roots I'm looking for. Using just yields as a root. Is it thrown out because a is supposed to represent the real part, but in that case it becomes the imaginary part? Or is it maybe because it represents the exact same roots as the other?