Quadratics in IB Mathematics: Analysis and Approaches centre on a small set of highly testable ideas: interpreting different forms, solving equations, using the discriminant, handling inequalities, and connecting algebra to parabolic graphs. The essential exam skill is not memorising one procedure, but recognising which representation or method reveals the required information most efficiently.
The current IB Mathematics: Analysis and Approaches guide places quadratic functions and equations within Topic 2: Functions. These ideas apply at both SL and HL, and they regularly appear inside longer questions involving intersections, parameters, modelling, or calculus.
The three forms of a quadratic
A quadratic function has the general form f(x) = ax² + bx + c, where a ≠ 0. Its graph is a parabola, opening upward when a > 0 and downward when a < 0.
IB questions expect you to move confidently among three forms:
| Form | Expression | Information visible immediately |
|---|---|---|
| Expanded form | f(x) = ax² + bx + c | y-intercept (0, c) and coefficients for the quadratic formula |
| Factorised form | f(x) = a(x − p)(x − q) | roots or x-intercepts x = p and x = q |
| Vertex form | f(x) = a(x − h)² + k | vertex (h, k), axis x = h, and maximum or minimum value |
For example,
f(x) = x² − 6x + 5 = (x − 1)(x − 5) = (x − 3)² − 4.
These equivalent forms reveal different features. The expanded form gives the y-intercept 5, the factorised form gives roots 1 and 5, and the vertex form gives the minimum point (3, −4).
Solving quadratic equations efficiently
Before solving, rearrange the equation into the form ax² + bx + c = 0. Then select a method based on the structure and wording of the question.
Factorisation
Use factorisation when integer or simple rational factors are visible. For example:
x² − 5x + 6 = 0
(x − 2)(x − 3) = 0, so x = 2 or x = 3.
The zero-product step matters. Writing only the factorised expression without stating both solutions does not complete the solution.
Completing the square
Completing the square is especially useful when the question asks for the vertex, axis of symmetry, range, or an expression in the form a(x − h)² + k.
For example:
x² − 6x + 5 = (x − 3)² − 9 + 5 = (x − 3)² − 4.
The vertex is therefore (3, −4) and the axis of symmetry is x = 3. Since the coefficient of the squared term is positive, the minimum value is −4.
The quadratic formula
For ax² + bx + c = 0, the solutions are
x = (−b ± √(b² − 4ac)) / 2a.
The formula is reliable when a quadratic does not factorise conveniently. Substitute with brackets, particularly when b or c is negative, and retain the ± until both solutions have been calculated.
The discriminant and the nature of the roots
The discriminant is
Δ = b² − 4ac.
It determines the number of real roots without requiring you to solve the equation completely.
| Discriminant | Real roots | Graphical meaning |
|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses the x-axis twice |
| Δ = 0 | Two equal real roots, also called a repeated root | Parabola touches the x-axis once |
| Δ < 0 | No real roots | Parabola does not meet the x-axis |
Exam questions often contain a parameter. If x² + kx + 9 = 0 has equal roots, then
k² − 4(1)(9) = 0, giving k² = 36 and therefore k = ±6.
Notice that both parameter values are required. A frequent mistake is taking only the positive square root.
Quadratic inequalities
A quadratic inequality asks where a parabola lies above or below the x-axis. First find the roots, then determine the sign in each interval.
For example:
x² − 5x + 6 ≤ 0
Factorising gives (x − 2)(x − 3) ≤ 0. The parabola opens upward, so it is on or below the axis between the roots, giving 2 ≤ x ≤ 3.
If the inequality were x² − 5x + 6 > 0, the answer would be x < 2 or x > 3. Strict inequalities exclude the roots, while ≤ and ≥ include them.
How IB examiners phrase quadratic questions
Recognising command language can save substantial time.
| Typical wording | Mathematical action |
|---|---|
| “Write in the form a(x − h)² + k” | Complete the square |
| “Hence state the minimum value” | Read k from vertex form when a > 0 |
| “Find the zeros” or “solve f(x) = 0” | Find the roots |
| “Determine the nature of the roots” | Calculate and interpret the discriminant |
| “Find the values of k for which there are two distinct roots” | Form Δ > 0 and solve the resulting inequality |
| “The line is tangent to the curve” | Set the equations equal and use Δ = 0 |
| “Solve graphically” | Use intersections or x-intercepts, normally with technology where permitted |
For a tangent problem, equate the line and parabola first. The resulting quadratic must have exactly one repeated real root, so its discriminant equals zero.
Exam technique and common errors
Quadratics award method marks as well as final-answer marks, so show enough algebra for the reasoning to be followed. In non-calculator work, an unsupported decimal answer may conceal the required method; where technology is permitted, still record the equation solved and interpret the output in context.
Common errors include:
- forgetting to rearrange the equation to zero before identifying a, b, and c;
- losing the negative sign in −b or −4ac;
- omitting one solution after using ±;
- giving roots when the question asks for coordinates, which should be written as
(x, 0); - treating the vertex y-coordinate as a minimum when a < 0, in which case it is a maximum;
- writing only the boundary points for an inequality instead of the required intervals;
- rounding intermediate values too early.
A useful checking routine is to substitute roots into the original equation, compare their midpoint with the axis of symmetry, and verify that the graph’s orientation agrees with the sign of a.
Turning knowledge into exam marks
Begin with the RevisionDojo quadratic function topic page, then practise targeted questions in the quadratics Questionbank. Attempt each question independently before studying the method, since passively reading solutions does not train method selection.
Use the Maths AA Functions video collection and the worked solutions attached to questions to see how algebra is organised line by line. Per-question past-paper video solutions are particularly useful for comparing your setup, notation, and calculator use with an efficient exam method.
Once individual skills are secure, mix quadratics with other functions through the broader Maths AA Questionbank. You can then use Maths AA predicted papers to practise recognising quadratics when the topic is not announced in advance. Jojo AI can help explain a missed step, but you should redo the question without assistance to confirm that the method is now independent.
Conclusion
IB Maths AA quadratics become manageable when you connect each form to the information it reveals. Factorised form exposes roots, vertex form exposes the turning point and range, expanded form supports formula and discriminant calculations, and the graph determines the solution of inequalities.
For effective revision, combine short algebra drills with complete exam questions and careful error analysis. RevisionDojo’s topic resources, Questionbank, Jojo AI explanations, and worked video solutions can help you move from understanding the theory to producing clear, mark-earning solutions under exam conditions.
Sources and referenced URLs
- Official IB Mathematics: Analysis and Approaches guide
- Official IB Mathematics: Analysis and Approaches subject brief
- Official IB Diploma Programme mathematics overview
- RevisionDojo IB Mathematics: Analysis and Approaches resources
- RevisionDojo quadratic function topic resources
- RevisionDojo quadratics Questionbank
- RevisionDojo Maths AA Questionbank
- RevisionDojo Maths AA Functions videos
- RevisionDojo Maths AA predicted papers