This is a reformulation of this question to better fit this forum. I removed the mentioning of sage math and this question is now 100% math. Given, the following quadrilateral: I want to describe the position of point X in terms of a multiple $t$ of the vector $HG$. The points $H$ and $G$ themselves […]

Let $az^2+bz+c=0$ be a quadratic equation with complex coefficients $a,b,c$ and roots $z_1, z_2.$ If it is given that $|z_1|\not=|z_2|,$ how can I obtain the condition for this containing $a,b,c?$ Is there any reference discuss about roots of quadratic equations with complex coefficients?

Is there an easier way to solve the following equation? $$x^2=2x-1$$ I think I know how to find $x$, using the quadratic formula: I get $$x^2-2x+1=0$$ then $$x=\frac{2 \pm \sqrt{4-4})}2= \frac{2 \pm \sqrt{0}}2$$ but I don’t know what $\sqrt{0}$ is. Is it $0$? If so, I would get $x=1$. Is that right? The teacher said […]

I have a problem with calculating the dual problem of : $$ \mbox{Minimize } tr(Y) + \frac{1}{\eta} tr(Z) $$ $$ \begin{pmatrix} Y & X \\ X & Z+\varepsilon I \end{pmatrix} \succeq 0 \mbox{, } % \begin{pmatrix} I & X \\ X & Z \end{pmatrix} \succeq 0$$ $$ \langle A_i,X\rangle = 0\mbox{, } i=1,\cdots,m \mbox{, } […]

Could you tell what are all the four points in following? Two circles intersect at two points maximum when we want to draw intersecting circles. But there we are solving quadratic equations, what is the argument about the other two missing points ?

I know how to find the solution for a quadratic equation with real coefficients. But if the coefficient changes to complex numbers then what is the change in the solution? Want an example of such equation with solution.

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