help with positive definite, semi definite, negative-definite, semi definite, thanks!

[amzzy]

Let a

and q be the quadratic form in three variables defined byq = ax₁² + 2ax₂² + (9a + 3)x₃² − 2ax₁x₂ − 6ax₁x₃ + 4ax₂x₃.




Find the values of a for which q is the following.
positive definite, positive semi definite, negative-definite and negative semi definite

 

[stapel]

Let a

and q be the quadratic form in three variables defined by q = ax₁² + 2ax₂² + (9a + 3)x₃² − 2ax₁x₂ − 6ax₁x₃ + 4ax₂x₃.

Find the values of a for which q is the following: positive definite, positive semi definite, negative-definite and negative semi definite

What are the definitions? How have you worked with this information? Where are you getting bogged down?

Please be complete. Thank you!