IB Maths: Quadratic Functions & the Discriminant — What You Actually Need to Know

You need three forms of a quadratic (expanded, factorised, vertex), the quadratic formula, and the discriminant $\Delta =b^2-4ac$ to determine the number of real roots. According to the IB Mathematics guide (first exams 2021, still current for AA/AI), this sits under Topic 2: Functions, tested at both SL and HL.

The discriminant $\Delta =b^2-4ac$ tells you how many real solutions a quadratic equation has. If $\Delta >0$, two distinct real roots. If $\Delta =0$, one repeated root (the parabola touches the x-axis). If $\Delta <0$, no real roots — the graph never crosses the x-axis.

Rearrange everything to one side first so it equals zero, then read off $a$, $b$ and $c$ before applying $\Delta =b^2-4ac$. This step trips up more students than the formula itself — especially when a line and curve are set equal to each other.

Completing the square rewrites $f(x)=a(x-h{)}^2+k$ directly, instantly giving you the vertex $(h,k)$ — useful for graph transformations and finding maximum/minimum values. The quadratic formula solves for roots numerically. Both derive from the same algebra, and IB examiners expect you to move fluently between them.

It appears on both. Paper 1 (no calculator, AA) tests the algebra directly — factorising, solving, discriminant conditions. Paper 2 (calculator allowed) tends to embed it inside longer context questions, like optimisation or geometry problems where a tangency condition needs $\Delta =0$.

Yes, but the depth differs. Analysis and Approaches treats quadratics as pure algebra — roots, graph properties, proof-style 'show that' questions. Applications and Interpretation uses quadratics mainly in modelling, so the discriminant is more of a practical check than a heavy algebraic focus.

The core discriminant content is identical at SL and HL — it's introduced once, in Topic 2. What changes at HL is how it's combined with harder material: complex roots when $\Delta <0$, polynomial division, and quadratics nested inside calculus or vector problems.

Discriminant-specific questions typically run 3-6 marks when standalone, but the concept regularly resurfaces inside longer 8-12 mark structured questions on Paper 2 — tangency conditions, optimisation, or finding a range of values for an unknown parameter.

The three I see every year: forgetting to rearrange the equation to equal zero before identifying $a$, $b$, $c$; mixing up the conditions for 'two distinct roots' versus 'real roots' (which includes the repeated case); and sign errors squaring a negative $b$.

Set up the discriminant inequality (usually $\Delta \ge 0$ or $\Delta >0$ depending on the wording), simplify to a quadratic in $k$, then solve that inequality — often needing its own discriminant or a sign diagram. Examiners award method marks even if your final inequality direction is wrong.

Work backwards from past paper questions rather than re-reading notes — the discriminant is a procedural skill, not a concept you can memorise passively. Mix pure algebra questions with modelling-style ones so you're comfortable spotting when a tangency or 'no solution' condition is hiding in context.

Not on its own. Quadratics and the discriminant appear in both subjects at similar difficulty in Topic 2 — the real difference between AA and AI is how algebra-heavy the rest of the course gets later, particularly calculus and proof. One shaky topic in Year 1 isn't a reliable signal for a subject switch.

Past paper questions grouped by topic work better than a textbook read-through, since the discriminant is tested as a procedure, not a definition to recall. Look for resources with worked solutions showing full method, not just final answers — that's where the marking-point habits actually get built.