I've been stuck on this for a while, can someone help me out? Let z be a complex number such that |z^2 + 4| = |z(z + 2i)|. Find the smallest possible value of |z + i|.
Note that z^2 + 4 = (z + 2i)(z - 2i), so we can write the given equation as |z + 2i||z - 2i| = |z||z + 2i|. If |z + 2i| = 0, then z = -2i, in which case |z + i| = |-i| = 1. Otherwise |z + 2i| /= 0, so we can divide both sides by |z + 2i|, to get |z - 2i| = |z|. This condition states that z is equidistant from the origin and 2i in the complex plane. Thus, z must lie on the perpendicular bisector of these complex numbers, which is the set of complex numbers where the imaginary part is 1. In other words z = x + i for some real number i. Then |z + i| = |x + 2i| = sqrt(x^2 + 4) >= 2. Therefore, the smallest possible value of |z + i| is 1, which occurs for z = -2i.