Solving a non-linear equation of complex numbers

Question

I have an equation as $$y=\beta_0|x|^2x + \beta_1x,$$ where $x$ is a complex number. I know $\beta_0$, $\beta_1$ and $y$. How can I get $x$ and solve the equation?

Many thanks.

Answer 1

Write $|x|^2 = x \bar x$. Then $$y=\beta_0 x^2 \bar x+\beta_1 x \tag 1$$

Conjugate to get

$$\bar y = \bar{\beta_0} \bar x^2 x+\bar{\beta_1}\bar x \tag 2$$

From $(1)$ solve for $\bar x$ and substitute in $(2)$. Simplify to get the cubic

$$\beta_0^2\bar y\ x^3+\beta_1(\beta_0\bar{\beta_1}-\bar{\beta_0}\beta_1)\ x^2+y(2\bar{\beta_0}\beta_1-\beta_0\bar{\beta_1})\ x-\bar{\beta_0}y^2=0$$

and use the known methods to solve cubic equations.