How to compute the dimension of the linear system of quadric hypersurfaces that pass through a given projective variety $S$?
A concrete one: $S$ is a $5$-dimensional projective variety in $\mathbb{P}^9$ which is the transverse intersection of $4$ quadric hypersurfaces.
Your question is equivalent to the computation of $H^0(\mathcal{I}_S(2))$.
In the example you give, $S$ is a complete intersection of $4$ quadrics and so the resolution of its ideal sheaf $\mathcal{I}_S$ is given by the Koszul complex (I write $\mathcal{O}$ instead of $\mathcal{O}_{\mathbb{P}^9}$):
$0 \to \mathcal{O}(-8) \to \mathcal{O}(-6)^{\oplus 4} \to \mathcal{O}(-4)^{\oplus 6} \to \mathcal{O}(-2)^{\oplus 4} \to \mathcal{I}_S \to 0$.
Tensoring with $\mathcal{O}(2)$ we obtain:
$0 \to \mathcal{O}(-6) \to \mathcal{O}(-4)^{\oplus 4} \to \mathcal{O}(-2)^{\oplus 6} \to \mathcal{O}^{\oplus 4} \to \mathcal{I}_S(2) \to 0$.
Splitting this exact sequence into short exact ones it is immediate to check that
$H^0(\mathcal{I}_S(2))=H^0(\mathcal{O}^{\oplus 4})=4$,
as Algori states in his comment.
Therefore the linear system of quadrics passing through $S$ has dimension $4-1=3$.
