I have a problem with determining eigenvalues. This is what I've got thus far:

Identify and sketch the graph of the quadratic equation

4x² + 10xy + 4y² = 9

This gives the matrix form:

Now we find the eigenvalues:

Det(A – xI) =

= x² – 8x – 9

= (x – 9)(x + 1)

eigenvalues are

From there, it's pretty simple solving:

My problem here is: How do I know which eigenvalue is which? It obviously makes quite a bit of difference to the final result. Nothing in my textbook says.