# Solving a non-linear equation of complex numbers

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.

• Can you provide more context? and specify the nature of $y, \beta_0$ and $\beta_1$ ? Oct 26, 2022 at 16:50
• $Y$, $\beta_0$ and $\beta_1$ are all scaler complex numbers. @NotaChoice Oct 26, 2022 at 18:03

## 1 Answer

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.