The discriminant

The bit under the root: The discriminant is Δ = b² − 4ac — the expression under the square root in the quadratic formula. Its sign tells you how many real roots the quadratic has, without solving it.

Sign of Δ → number of roots: Δ > 0 → two distinct real roots (graph cuts the x-axis twice). Δ = 0 → one repeated root (graph touches the x-axis). Δ < 0 → no real roots (graph misses the x-axis).

You don't need to solve: If a question only asks 'how many solutions' or 'show it has no real roots', just compute Δ and read its sign.

Touching = one solution = Δ = 0: A line is tangent to a parabola when they meet exactly once. Set them equal, form a quadratic, and set its discriminant to 0 — that's the touching condition.

Two meet → discriminant: Whenever a line and a curve meet a specific number of times, equate them, form a quadratic, and use Δ: =0 (tangent), >0 (two points), <0 (never).

Turn the root-condition into an equation in the unknown: "Equal roots" → Δ = 0; "two distinct roots" → Δ > 0; "no real roots" → Δ < 0. Write Δ in terms of the unknown and solve the resulting equation or inequality.

Read the condition carefully: "Two different solutions" is Δ > 0 (an inequality); "a repeated/equal root" is Δ = 0 (an equation). Match the wording.