Hi.

So, here's the set up. I have 3 equations:

$\displaystyle \displaystyle A = a\xi_x^2 + b\xi_x \xi_y + c \xi_y^2 $

$\displaystyle \displaystyle B = 2a\xi_x \eta_x + b(\xi_x \eta_y + \xi_y \eta_x) + 2c\xi_y\eta_y $

$\displaystyle \displaystyle C = a\eta_x^2 + b\eta_x \eta_y + c \eta_y^2 $

And my task is to show that:

$\displaystyle \displaystyle B^2-4AC = (b^2-4ac)(\xi_x \eta_y - \xi_y \eta_x)^2 $

I can give the same equations using alternative variables if it makes it easier on the eyes for you:

$\displaystyle \displaystyle A = aw^2 + bw x + c x^2 $

$\displaystyle \displaystyle B = 2aw y + b(w z + x y) + 2cxz $

$\displaystyle \displaystyle C = ay^2 + by z + c z^2 $

Show that:

$\displaystyle \displaystyle B^2-4AC = (b^2-4ac)(w z - x y)^2 $

My Question

Is there a way to do this that doesn't involve actually having to square B, and muliply A and C, leading to a disturbingly large number of terms in the equation?

I feel like there should be a way to exploit the symmetry of the equations to create factors which allow me to deduce the final result in a vastly reduced number steps with an appropriate number of terms in my derivation.

I should stress that this is not a homework piece, but a small part of a much larger derivation that I am doing for practice, and wonder if it can be economised.