I have the equation so that $x^2+x+2=0$. Since $\Delta<0$ we have two complex conjugate solutions. I am trying to visualise them on the Argand-Gauss plane to understand the algebra.
I have three axes such as:
- The $X$-axis for the real numbers (input)
- The $Y$-axis for the pure imaginary numbers (input)
- The $Z$-axis for the real numbers (output)
Therefore, the input is a complex number $x_c=(x+yi)$.
With the equation $x^2+x+2=0$, I will write a complex function so that:
$$ z={x_c}^2+x_c+2\\ z={(x+yi)}^2+(x+yi)+2\\ z=x^2+2xyi-y^2+x+yi+2\\ z=\color{green}{(x^2-y^2+x+2)}+\color{red}{(2xy+y)i} $$
I have the real function $\color{green}{f(x,y)}$ in green and the pure imaginary function $\color{red}{g(x,y)}$ in red. I decided to plot these two functions in GeoGebra 3D:
Correct me if I am wrong, but I believe the output of $\color{red}{g(x,y)}$ has no tangible value. It outputs pure imaginary numbers that cannot be shown.
The 2D parabola is shown here:
Naturally, the complex solutions are the intercepts of the real and imaginary part:
This is the parabola known as the "phantom graph", its maximum is the minimum of the real parabola and the intercepts on the Argand-Gauss plane are the solutions:
I would like to determine the expression of the "phantom graph", or at least, demonstrate it. I have seen this GeoGebra activity where the "phantom graph" is determined with the expression:
$$ z=-a{x_c}^2+c-\frac{b^2}{4a} $$
${x_c} = (x+yi)$ with the real part fixed to a certain offset (here $-\frac{1}{2}$).
Contrary to the real parabola which is:
$$ z=ax^2+2x+c $$
I would like to demonstrate the expression $z=-ay^2+c-\frac{b^2}{4a}$ and understand why this "phantom graph" turned upside down and rotated 90 degrees along the $Z$-axis yields the solutions when it intercepts the Argand-Gauss plane.