Quadratic forms, linear algebra

[matrix_204]

I have a question that i have to do, the only problem is time. Since i have to finish my stats assignment, would anyone tell me the steps involved in solving this problem(in order),
Rotate and translate the coordinate axes, as necessary to bring the conic section
3x^2 -8xy -12y^2 -30x-64y=0
into standard position. Give its equation in standard form.
Sketch it in relation to original and final axes.

This problem is not difficult for me to do it, but it just takes me time to write it all up, and if anyone could just tell me the steps in order to fully solve this, it would really help me, timewise, thanks.

 

[Galileo]

How I'd do it:

First complete the squares so you can see what translation you should make to bring it into a quadratic form.
Then write down the symmetric coefficient matrix and diagonalize it.
See to what rotation the diagonalizing orthogonal matrix corresponds.
Classify the quadratic form and sketch.

 

[HallsofIvy]

What do you mean it "is not difficult" but takes too long to write up? If you can do the problem why do you need us to tell you what to do?

I don't see any reason to complete the square first: Just write the matrix for the
quadratic terms:

\left[\begin{array}{cc}3&-4\\-4&-12\end{array}\right].

(Notice that the "-8" of "-8xy" is divided between the two non-diagonal terms, making the matrix symmetric so it can be diagonalized.)

Find the eigenvalues and eigen vectors. Choosing the eigenvectors as new axes makes the new coefficient matrix diagonal- i.e. removes the xy term.

 

[matrix_204]

i understand the problem, but as i said timewise i was short, whether its difficult or not i didn't have time to do this while doing my stats assignment, but anyways i got it now, thnx