I am also familiar with Vieta from math competitions but I think this is more than that.

Given two numbers, r1 and r2, if you know their arithmetic mean:

  m = (r1 + r2) / 2

and geometric mean:

  g = sqrt(r1 * r2)

then following Loh's derivation you get a very cute formula:

  r1 = m - sqrt(m^2 - g^2)

  r2 = m + sqrt(m^2 - g^2)

Vieta gives an easy way of finding those means from a quadratic equation: r1+r2=-b and r1*r2=c. So you can plug in m=-b/2 and g=sqrt(c) in the equation above.

The fact that you can state roots in terms of their means is the more novel insight to me. (note: Loh doesn't talk about geometric means but I thought using just product of the roots isn't as "symmetric")