Symmetric Matrices & Quadratic Forms · Hard problem

A structure's strain energy is a quadratic form $q(\mathbf x)=\mathbf x^{T}K\mathbf x$ for its (symmetric) stiffness matrix $K$. Two displacement modes $\mathbf u$ and $\mathbf v$ have energies $q(\mathbf u)=10$ and $q(\mathbf v)=14$. When both modes are excited together, the combined displacement $\mathbf u+\mathbf v$ has energy $q(\mathbf u+\mathbf v)=40$. Find the interaction energy $\mathbf u^{T}K\mathbf v$, without ever learning $K$, $\mathbf u$, or $\mathbf v$ explicitly.