Find the two square roots of 5-2i in the form a+bi, where a and b are real.

Question

Mark on an Argand diagram the points P and Q representing the square roots. Find the complex number of R and S such that PQR and PQS are equilateral triangles.
How do you do the last part finding R and S??

Answer 1

well, I understand the first part
sqrt (5-2i)
z^2 = (x+iy)^2 = 5 - 2 i
then
(x^2-y^2) + 2 x y i = 5 - 2 i
so
x^2 - y^2 = 5
2 x y = -2 (note x and y opposite signs, helps later)
the magnitude of 5 - 2 i squared = 25 + 4 = 29 = x^2 + y^2
so now two equations
x^2 - y^2 = 5
x^2 + y^2 = 29
--------------------- add
2 x^2 = 34
x^2 = 17
x = + or - sqrt 17
if x = + sqrt 17, then y = -sqrt 12
if x = - sqrt 17, then y = +sqrt 12
sqrt 17- i sqrt 12
and
-sqrt 17 + i sqrt 12