I've been stuck on this for a while, can someone help me out? Let ** z** be a complex number such that

NewMember Dec 1, 2019

#3**+5 **

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**.

NewMember Dec 3, 2019