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Complex Squares

Stage: 5 Challenge Level: Challenge Level:1

What do you get if you square $a+ib$?

Working backwards:

if $(a+ib)^2$ is real, what relationships must $a$ and $b$ satisfy?
if $(a+ib)^2$ is imaginary, what relationships must $a$ and $b$ satisfy?

What does this look like on an Argand diagram?