Key Idea: Quadratics turn up everywhere in Unit 2 — you'll solve them, count their real roots with the discriminant, and read off where they're positive or negative. It's mostly Paper 1 (by hand), with the GDC as a Paper-2 shortcut.
🔢 First: set it to = 0, then choose a method
- read off ax² + bx + c = 0 — equation set to zero first
- the two roots — keep both signs
- the bit under the root: the discriminant Δ
| When the question… | Method | Key move |
|---|---|---|
| factorises nicely | Factorise into two brackets | Set each bracket = 0: (x−2)(x−3)=0 ⇒ x = 2 or 3. |
| is messy / 'give to 3 s.f.' | Quadratic formula | Bracket every substitution; a negative c makes −4ac positive. |
| asks 'in the form (x−h)²' | Complete the square | Square-root both sides — keep the ± or you lose a root. |
| is on Paper 2 / awkward numbers | GDC (solver or graph) | Read the x-intercepts; always allowed on Paper 2. |
🔎 The discriminant: Δ = b² − 4ac
- the discriminant — the sign counts the real roots
- from ax² + bx + c = 0 (set to zero first)
| Discriminant | Real roots | What the graph does |
|---|---|---|
| Δ > 0 | two distinct real roots | cuts the x-axis twice |
| Δ = 0 | one repeated (equal) root | touches the x-axis (tangent) |
| Δ < 0 | no real roots | misses the x-axis entirely |
'Equal roots' → Δ = 0 (an equation); 'two distinct roots' → Δ > 0; 'no real roots' → Δ < 0. A line is tangent to a parabola when they meet once — set them equal, form a quadratic, use Δ = 0. To solve a quadratic inequality: get one side to = 0, find the roots, then let the parabola decide. Upward (a > 0): negative BETWEEN the roots, positive OUTSIDE. Write 'between' as one chain (p ≤ x ≤ q), 'outside' as two pieces joined by or.
✏️ IB-style worked examples
IB-style question — solve with the quadratic formula (Paper 1)
Solve 3x² + 5x − 1 = 0, giving your answers to 3 s.f.
Step by step:
Read off a, b, c and substitute into the formula.
Simplify under the root — the −4ac becomes +12.
Take each sign separately.
x = 0.180 or x = −1.85 (3 s.f.).
IB-style question — find k for equal roots, Δ = 0 (Paper 1)
The equation x² + (k+1)x + 4 = 0 has equal roots. Find the possible values of k.
Step by step:
Equal roots ⇒ Δ = 0. Here a = 1, b = k + 1, c = 4.
Expand and tidy to a quadratic in k.
Factorise and solve.
k = −5 or k = 3 (either value gives a repeated root).
IB-style question — solve a quadratic inequality (Paper 1)
Solve x² − 2x − 8 > 0.
Step by step:
Find the roots: solve x² − 2x − 8 = 0.
a = 1 > 0 (upward U); '> 0' means the OUTSIDE regions.
x < −2 or x > 4 (two pieces, joined by 'or').
🔒 GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Important: Quadratics almost always have two answers. Keep the ± in the formula and when square-rooting (√9 = 3, but (x−4)² = 9 needs ±3). Watch the signs in −4ac: a negative c makes it positive. And for an inequality, decide between vs outside from the parabola — don't just write the roots.
Tap each card to reveal the answer.
Solve x² − 5x + 6 = 0 by factorising x = 2 or x = 3 — two numbers that multiply to 6, add to −5.
Solve (x − 4)² = 9 x = 7 or x = 1 — square-root both sides: x − 4 = ±3.
How many real roots has 2x² − 4x + 1 = 0? Two — Δ = (−4)² − 4(2)(1) = 8 > 0.
x² + kx + 9 = 0 has equal roots — find positive k k = 6 — Δ = 0 ⇒ k² − 36 = 0 ⇒ k = ±6.
For which c is y = x + c tangent to y = x²? c = −1/4 — x² − x − c = 0 with Δ = 0 ⇒ 1 + 4c = 0.
Solve x² − x − 6 ≤ 0 −2 ≤ x ≤ 3 — upward parabola, '≤ 0' is between the roots (ends included).
Exam Tips
- Always rearrange to = 0 first, then pick: factorise (clean), formula (always works), complete the square (when asked), GDC (Paper 2).
- Quadratic formula is in the booklet — bracket every substitution and watch that −4ac flips sign with a negative c.
- Δ = b² − 4ac counts roots: > 0 two, = 0 one (touching/tangent), < 0 none — no need to solve.
- Root-condition → equation/inequality in the unknown: equal roots ⇒ Δ = 0; tangent line ⇒ Δ = 0.
- Inequalities: find the roots, then read off the parabola — negative between, positive outside (upward); 'outside' uses 'or'.