Discriminant conditions: tangents, and "does not cross"

The discriminant is not just "how many roots" — it is the tool for tangency and range-of-values questions too.

Translating the condition into a discriminant equation

A line is TANGENT to a curve exactly when substituting the line into the curve's equation gives a quadratic with a REPEATED root — set discriminant = 0 and solve for the unknown.
A quadratic "does NOT cross the x-axis" (has no real roots) exactly when its discriminant is LESS than zero — set b² − 4ac < 0 and solve the resulting inequality for the unknown constant.
A quadratic that DOES cross the x-axis twice needs discriminant > 0 — the same setup, opposite inequality direction.
Always substitute FIRST (the line into the curve, or the given values into a, b, c), THEN apply the discriminant condition — applying the condition before substituting is the most common setup error.

Worked example Finding k where a line is tangent to a quadratic The line 2y = 2x + 29 is tangent to x² − 2ky + 32k = 0. Find k. Rearrange the line to y = x + 14.5, then substitute into the quadratic: x² − 2k(x + 14.5) + 32k = 0. Expand: x² − 2kx − 29k + 32k = 0 → x² − 2kx + 3k = 0. TANGENT means a repeated root, so the discriminant of this quadratic in x equals zero: (−2k)² − 4(1)(3k) = 0 → 4k² − 12k = 0. Factorise: 4k(k − 3) = 0 → k = 0 or k = 3. Check which value makes sense in context (k = 0 may make the original equation degenerate, so k = 3 is the answer expected). ✓ Substituting the line into the curve BEFORE applying the discriminant condition is the essential first step — the discriminant only applies to the resulting single-variable quadratic. ✕

Using the wrong inequality direction for "does not cross"

"Does not cross the x-axis" means NO real roots, which is discriminant < 0 (strictly less than) — not ≤ 0. Confusing "does not cross" (< 0) with "touches or does not cross" (≤ 0) changes whether the boundary value itself is included in the answer.

Discriminant value	What it means
b² − 4ac > 0	Two distinct real roots — crosses the x-axis twice
b² − 4ac = 0	One repeated root — tangent / touches the x-axis once
b² − 4ac < 0	No real roots — never crosses or touches the x-axis

Practice for this standard covers the whole topic — flashcards and quiz.