# how to solve a quadratic equation in three unknowns

Is there any elegant way of solving this system of equations?

\begin{gather*} a_1 x^2+a_2 x+ a_3 y^2+a_4 y+a_5 z^2+a_6 z+a_7=0 \\
b_1 x^2+b_2 x+b_3 y^2+b_4 y+b_5 z^2+b_6 z+b_7=0\\
c_1 x^2+c_2 x+c_3 y^2+c_4 y+c_5 z^2+c_6 z+c_7=0 \end{gather*}

Still, any solution will be appreciable.

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Hmm... intersecting ellipsoids? Where did your nonlinear system come from? –  Ｊ. Ｍ. Apr 20 '11 at 8:25
@J.M. first i would like to know how to put a1, secondly, it comes from circle in 3-d ... generalized to polynomials.. lets say a first step of the problem towards solution. –  Santosh Linkha Apr 20 '11 at 8:29
Could you elaborate on the application where it came up? We can either bash at the problem with Yuval's suggestion, or study what led you to your equation set and probably tease out a way to solve this without heavy machinery... –  Ｊ. Ｍ. Apr 20 '11 at 8:32
@J.M. well, currently comes from stackoverflow.com/questions/5712963/… but still,it's an old problem of mine. .... still i would be more happy to solve it for myself than help him. –  Santosh Linkha Apr 20 '11 at 8:33
So you want the circumcenter of a tetrahedron? That is a much easier problem than what you've presented. It certainly won't need Gröbner. –  Ｊ. Ｍ. Apr 20 '11 at 8:36