Mathematics
Elementary Symmetric Polynomials and Vieta's Formulas
Quick fact
Viète's formulas show that for a monic polynomial, each coefficient is exactly the elementary symmetric polynomial of the roots, up to a sign. For quadratics, this gives the sum and product of roots directly from the coefficients.
Why this is interesting
You probably know that the sum of the roots of a quadratic is -b/a. But did you know that every coefficient of a polynomial secretly encodes a symmetric pattern of its roots?