Let $\mathbf{x}=(x_1,x_2,x_3)^T$. You may first find a symmetric matrix $A$ such that $q(x_1,x_2,x_3)=\mathbf{x}^TA\mathbf{x}$. Since $A$ is symmetric, it is orthogonally diagonalizable. So, find three unit eigenvectors $u,v,w$ such that correspond to the three eigenvalues of $A$. Then $Q=[u,v,w]$ will automatically be an orthogonal matrix and $A=QDQ^T$, where $D$ is the diagonal matrix whose diagonal entries are the eigenvalues of $A$. Now, let $\mathbf{z}=(z_1,z_2,z_3)^T = Q^T\mathbf{x}$. Express $q(x_1,x_2,x_3)$ in terms of $z_1,z_2,z_3$. What do you see? If you have carried out the above steps correctly, by now you shall be very close to the solution.