Mix of Z and |Z| - Part 2

$\begin{aligned} Z^3 + |Z|^2 + Z & = 0 \\ Z^3 + Z\bar Z + Z & = 0 & \small \color{#3D99F6} \text{where }\bar z \text{ is the conjugate of }z. \\ Z\left(Z^2 + \bar Z + 1\right) & = 0 & \small \color{#3D99F6} \text{Since }Z = a + jb \ne 0, \\ Z^2 + \bar Z + 1 & = 0 \\ a^2 - b^2 + j2ab + a - jb + 1 & = 0 \end{aligned}$

Equating the real and imaginary parts on both sides:

$\begin{cases} a^2 - b^2 + a + 1 = 0 & ...(1) \\ 2ab - b = 0 & ...(2) \end{cases}$

From $(2): \ 2ab - b = b(2a-1) = 0$ , since $b \ne 0$ , $\implies a = \dfrac 12$ . From $(1): \ \frac 14 - b^2 + \frac 12 + 1 = 0$ , $\implies b = \dfrac {\sqrt 7}2$ . Therefore $a + b = \dfrac {1+\sqrt 7}2 \approx \boxed{1.823}$ .