The discriminant: which inequality, and which direction

5 min read·Updated 5 September 2026

For a quadratic equation $ax^2 + bx + c = 0$ with $a \ne 0$, the discriminant

decides the nature of the roots and nothing else. It is one of the shortest pieces of theory in the syllabus, which is exactly why questions about it are built on the fine print rather than the idea.

The three cases:

• $\Delta \gt 0$ — two distinct real roots;
• $\Delta = 0$ — one repeated (double) real root;
• $\Delta \lt 0$ — no real roots.

Misconception 1: the wrong strictness

"Two real roots" and "two distinct real roots" are different conditions. The first is satisfied by $\Delta \ge 0$ — a repeated root is usually counted as two equal real roots — while the second demands $\Delta \gt 0$. Under time pressure the word distinct is the easiest word on the page to skim past, and skipping it changes the boundary of your answer without changing anything else about your working.

The mirror image is just as common: "the graph touches the x-axis" means $\Delta = 0$ exactly, not $\Delta \ge 0$.

Misconception 2: losing a branch when you take a square root

A condition like $k^2 \gt 36$ has two branches: $k \gt 6$ and $k \lt -6$. Students who solve it as if it were a linear inequality write only the positive branch. The answer is not wrong so much as half-present, and a half-present answer is exactly what a well-built option list offers you.

The safe move is to factorise rather than square-root: $k^2 - 36 \gt 0$ becomes $(k-6)(k+6) \gt 0$, and a product of two factors is positive when both are positive or both are negative — which produces both branches by construction.

Misconception 3: pattern-matching a factorisation

A difference of two squares, $x^2 - 9$, factorises. A sum of two squares, $x^2 + 9$, does not (over the real numbers). Expressions that merely resemble a standard identity get factorised as if they matched it, and the resulting "roots" are numbers that solve a different equation.

The related habit is stopping one step early: after pulling out a common factor, check whether what remains still factorises.

A worked example

For what values of $k$ does $x^2 + kx + 9 = 0$ have two distinct real roots?

Here $a = 1$, $b = k$, $c = 9$, so

Distinct means strictly positive:

Two wrong turns, both of which land on a plausible option. Reading distinct as merely real gives $k \le -6$ or $k \ge 6$, which differs from the correct answer only at two points — and those two points are precisely where the equation has a repeated root, so the difference is the whole content of the question. Solving $k^2 \gt 36$ by "taking the square root" gives $k \gt 6$ alone, silently discarding every negative value of $k$ that works.

Check yourself

Two words before you commit: "distinct?" and "both branches?" If the question contained distinct, equal, touches or no real roots, translate that word into one of the three cases explicitly, in writing, before you solve anything.

The drill below varies the coefficient the condition is placed on, so the translation has to be done again each time.

Original study notes by Dsetopia, written from the EDB syllabus learning objectives. Not affiliated with the HKEAA; no exam content is reproduced.