Nonlinear Equations & Functions at the 700-800 Level

Four archetypes generate nearly every hard item here. Tangency via the discriminant: "intersect at exactly one point" means set the expressions equal and force b² - 4ac = 0 — but the parameter you solve for is rarely the final answer; the item asks for the tangent point's coordinates or filters the parameter by sign, and the full trail of intermediates is planted as distractors. Radical equations: squaring manufactures extraneous roots; the fast filter is that the non-radical side must be nonnegative, which identifies the doomed root before any substitution. Line–parabola intersections: Vieta hands you the sum of x-coordinates in one step, so the test asks about y-coordinates — push each x through the line, not the parabola. Horizontal-line problems ("two points d apart") collapse under symmetry: find the axis, step d/2 to each side, evaluate once. Absolute-value models: isolate |x - c|, remember the inequality flips when dividing by a negative, and keep half-width, endpoints, interval length, and integer count straight — all four numbers appear among the choices, and the integer count needs the +1.

The module drills the pre-checks and shortcuts that make these 75-second problems instead of 3-minute ones, plus the Desmos moves (graph both sides; extraneous roots never appear) that verify answers instantly.