your quadratic form can be written as a matrix multiplication as

We can diagonalize that matrix by finding its eigenvalues and eigenvectors. Its characteristic equation is

You can find the eigenvalues by solving that equation using the quadratic formula. They are not, as already pointed out, "easy" numbers: something like, I think

. Find the eigenvectors corresponding to that and construct the matrix P having those eigenvectors as columns.

will be the diagonal matrix having the eigenvallues as entries.