How are the discriminant in a single variable quadratic equation and the binary quadratic form over the unit circle related?
I will point out that the discriminant in the quadratic formula has the opposite sign from the used in the following.
This is a screen scrape from my notes based on Edwards showing the determination of the Lagrange multipliers for general binary quadratic form.
The symbol $\mathcal{J}$ represents a rotation by $\pi/2$ and the expression where it appears is a standard dot product.
If the discriminant is $ac-b^2\ne0$ is positive then both extremal values have the same sign. Otherwise they have different signs.
This is the graph for a case of a negative discriminant.
In a quadratic equation in one variable, the discriminant tells us whether we have two real solutions, a single solution or only imaginary solutions. It seems to me, these two situations must be closely related.
The discriminate of the general quadratic equation is invariant under orthogonal transformations. It can also be used to eliminate the cross $xy$ term in the case of two variables.
Addendum:
If someone takes the time to present this in proper mathematical statements, I'll be happy to accept the answer. The connection is "obvious". Imagine rotating a cutting plane containing the $z$ axis. At each angle the intersection between the graph of the quadratic surface and the cutting plane will be a (possibly degenerate) parabola. If these parabolae all lie entirely above or below the $x\times{y}$ plane, then the corresponding quadratic formulae have no real solutions. This is the same thing as saying the graph lies entirely above or entirely below the $x\times{y}$ plane.