# geometric interpretation of quadratic equation with complex coefficients

When an equation has real coefficients and non-negative discriminant, the geometric meaning of it's roots is intersection of the function with the x-axis.

I know how to get roots of quadratic equation with complex coefficients, I just wonder if there is any geometric interpretation for that.

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Try my paper, Graphing the Complex Roots of Quadratic Functions on a Three Dimensional Coordinate Space, detailing a geometric construction detailing how to derive the complex roots geometrically.

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Say you have a polynomial $F(x_1,...,x_n)$ with real or complex coefficient. Then you can consider it as definining a function $$K^n\longrightarrow K\qquad\qquad(*)$$ where $K$ is either $\Bbb R$ or $\Bbb C$. Simply, the value associated to an $n$-ple $\vec a=(a_1,...,a_n)$ is $F$ evaluated at $\vec a$. In geometric terms, solving the equation $$F(x_1,...,x_n)=0$$ means finding the zero-locus of the function $(*)$ which is some subset $Z_F\subseteq K^n$. The case you are familiar with is that of $n=1$, where $Z_F$ is typically a set of isolated real or complex points.

Studying the geometry of the sets $Z_F$ is the foundational motivation of Algebraic Geometry

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Thanks for your response, but i'd like to know if it is possible to draw the roots and see how they look like. Wikipedia article doesn't answer my question. – koddo Feb 21 '13 at 14:49
Since you talk about "roots", I'm assuming that you're looking at the one variable case. Sometimes the roots can be computed exactly, more often not. In any event, there techniques that allow to find approximations good enough to make an accurate drawing. – Andrea Mori Feb 21 '13 at 17:17
Excuse me, but you do not need to assume anything since i explicitly wrote i'm interested in quadratic equation. – koddo Feb 21 '13 at 18:06
quadratic polynomials in one variable always admit two, possibly equal, complex roots. In more variables their zero-locus are called quadrics which, in many respect, are analogous to the conic sections first studied by the Greeks, e.g. see en.wikipedia.org/wiki/Quadric – Andrea Mori Feb 21 '13 at 21:54