Quadratic Equations at the 700-800 Level

Hard-tier quadratic items almost never say "solve this equation." They are built from three archetypes. Discriminant with a parameter: a line is tangent to a parabola (or an equation has no real solutions) and an unknown constant controls it — set the expressions equal, write $b^2 - 4ac$ in terms of the parameter, and set it to 0, > 0, or < 0. The twist is that the question usually asks for a value downstream of the parameter (the tangent point's y-coordinate, say), and every intermediate value is planted as a wrong answer. Vertex optimization with a hidden step: max-height and max-revenue models where you must build the quadratic yourself from "each \2$ increase loses 10 members," convert units ("x hundred units"), or respect a whole-number restriction that makes the exact vertex value unattainable. Vieta shortcuts: sum and product of roots answer questions about 1/r + 1/s or r² + s² in ten seconds with no radicals.

The distractors are systematic: the intermediate value, the unit slip, the sign flip, and the unattainable vertex. This module drills recognizing the archetype fast, taking the Vieta/symmetry route instead of the quadratic formula, and always re-reading what quantity the final sentence actually requests.