Solution to a quadratic form I'm trying to find a closed form solution of the following quadratic form for $x$.
$x^{T}Dx = c$
where $c$ is just a constant placeholder for some terms on the other side.
I know that, because $D$ is positive semi-definite, this equation describes a level curve of an ellipsoid in $x$, which gives me hope that there exists a closed form solution of the form $x = \frac{a\pm  \sqrt{z}}{b}$, for some $a, z, b$ (which may be matrices or vectors). 
Do you know if this solution exists and if so how it is determined? 

Write this in a basis with respect to which $D$ is diagonal.  Then I think you can see better what is going on.  You can do this using a basis of orthogonal vectors.  Also, if $D$ is positive definite, you might find the Cholesky factorization less computationally intensive.
