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.
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2$\begingroup$ Can you provide more context? and specify the nature of $y, \beta_0$ and $\beta_1$ ? $\endgroup$– NotaChoiceOct 26, 2022 at 16:50
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$\begingroup$ $Y$, $\beta_0$ and $\beta_1$ are all scaler complex numbers. @NotaChoice $\endgroup$– zahraOct 26, 2022 at 18:03
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1 Answer
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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.
