Quadratics in IB Mathematics: Analysis and Approaches are examined through a small set of recurring structures: solving equations, interpreting graphs, using the discriminant, finding unknown parameters, solving inequalities, and building models. The context may change, but the underlying methods are highly predictable.
To answer IB Maths AA quadratics questions, first identify what the question is really asking, choose the form of the quadratic that exposes the required information, and show enough working to justify the result. The most effective revision method is to attempt a question independently, watch or study its complete worked solution, and then solve a similar question without assistance.
What IB Maths AA requires you to know about quadratics
The official Mathematics: analysis and approaches guide places quadratic functions within the functions topic. At both SL and HL, students are expected to understand the graph of a quadratic, its intercepts, axis of symmetry, vertex, roots, discriminant, and different algebraic forms.
The essential forms are:
Form
Expression
Information immediately visible
Standard form
f(x)=
4.3
X
Share on WhatsApp
Share on LinkedIn
Share on Facebook
a
x2
+
bx+
c
y-intercept (0,c) and coefficients for the quadratic formula
Factored form
f(x)=a(x−p)(x−q)
Roots x=p and x=q
Vertex form
f(x)=a(x−h)2+k
Vertex (h,k) and axis of symmetry x=h
In every form, a=0. If a>0, the parabola opens upwards and has a minimum. If a<0, it opens downwards and has a maximum.
The syllabus also includes solving quadratic equations by factorization, completing the square, and the quadratic formula. Students must use the discriminant
Δ=b2−4ac
to determine the nature of the roots:
Δ>0: two distinct real roots
Δ=0: two equal real roots, often described as one repeated root
Δ<0: no real roots
For HL students, a negative discriminant can lead to complex solutions, but the statement “no real roots” remains important when interpreting a real graph or physical model.
The main types of quadratics questions
Solving a quadratic equation
Suppose you are asked to solve
2x2−5x−3=0.
Always set the equation equal to zero before factorizing:
2x2−6x+x−3=02x(x−3)+1(x−3)=0(2x+1)(x−3)=0.
Therefore,
x=−21orx=3.
Factoring is fastest when the factors are reasonably clear. If they are not, use the quadratic formula rather than spending excessive time guessing.
For ax2+bx+c=0,
x=2a−b±b2−4ac.
A frequent error is treating the denominator as 2 rather than 2a. Substituting the values of a, b, and c into a separate line reduces sign errors.
Finding a vertex or maximum or minimum
Consider
f(x)=2x2−8x+5.
Complete the square:
f(x)=2(x2−4x)+5=2[(x−2)2−4]+5=2(x−2)2−3.
The vertex is therefore (2,−3), the axis of symmetry is x=2, and the minimum value is −3. Notice the distinction between the point (2,−3) and the minimum value −3.
You can also find the vertex’s x-coordinate using x=−b/(2a) and then substitute it into the function. Completing the square is usually preferable when the question asks you to express the function in vertex form or use a result in a later part.
Using the discriminant to find a parameter
A common IB structure introduces an unknown constant and asks when a quadratic has one real root. For example, find k if
x2+kx+9=0
has two equal real roots.
Equal roots mean that the discriminant is zero:
k2−4(1)(9)=0k2=36k=6ork=−6.
Do not stop at k2=36 and give only k=6. Parameter questions frequently produce two valid values.
The wording controls the discriminant condition:
Wording
Required condition
Two distinct real roots
Δ>0
Two equal real roots or tangent to the x-axis
Δ=0
At least one real root
Δ≥0
No real roots
Δ<0
Solving a quadratic inequality
To solve
x2−x−6<0,
first find the boundary values:
(x−3)(x+2)=0,
so the roots are −2 and 3. Because the coefficient of x2 is positive, the graph opens upwards and lies below the x-axis between its roots. Therefore,
−2<x<3.
The strict inequality means the endpoints are excluded. For ≤0, the answer would be −2≤x≤3.
A sign diagram is safer than memorizing “inside” or “outside.” If the leading coefficient is negative, the sign pattern reverses. Always test one value in each interval if the graph’s sign is not immediately clear.
Forming a quadratic from given information
Suppose a quadratic has roots 2 and −5 and passes through (1,−18). Start with its factored form:
f(x)=a(x−2)(x+5).
Substitute the known point:
−18=a(1−2)(1+5)=−6a,
so a=3. Hence,
f(x)=3(x−2)(x+5).
A common mistake is assuming a=1. Roots determine the factors, but an additional point or coefficient is normally needed to determine the vertical scale.
Quadratics in modelling questions
Quadratics may describe height, area, revenue, or profit. These questions test interpretation as well as algebra. A model such as
h(t)=−5t2+20t+1
may be mathematically defined for every real t, but time in the context will normally require t≥0. A negative solution may therefore need to be rejected.
When interpreting a model, state units and answer the contextual question. If the vertex occurs at (2,21), write that the object reaches a maximum height of 21 metres after 2 seconds, rather than reporting only two unexplained numbers.
A reliable method for any quadratics question
Use this sequence under exam conditions:
Identify the target. Are you finding roots, a vertex, a parameter, an interval, or a modelled quantity?
Rewrite if necessary. Put an equation equal to zero or convert the function into a more useful form.
Choose the method. Use factoring for simple roots, completing the square for a vertex, the discriminant for the nature of roots, and graphing technology for permitted numerical work.
Show substitution and key algebra. A correct unsupported number may not demonstrate the required method.
Check all solutions. Look for a missing ±, excluded endpoints, domain restrictions, or extraneous contextual answers.
Present the requested form. Give exact values unless a decimal is requested or appropriate, and include units in applications.
The IB defines find and calculate as requiring relevant stages of working. Hence tells you to use the preceding result, while hence or otherwise permits another valid method. Paying attention to these command terms helps you avoid solving a question in a longer or less relevant way.
Paper 1 and Paper 2 strategy
According to the official IB Mathematics AA subject brief, Paper 1 does not allow technology, while Paper 2 requires technology. HL students also sit the technology-required Paper 3.
On Paper 1, expect quadratics to reward clean algebra, exact values, completing the square, factorization, or discriminant reasoning. Do not turn an exact answer such as (3+5)/2 into an unnecessary decimal.
On Paper 2, a graphing display calculator can find intersections, roots, and turning points numerically. However, calculator output does not replace reasoning when the question asks you to show, justify, or determine a parameter algebraically. Enter the function carefully, select a sensible viewing window, and record the mathematical result rather than calculator keystrokes.
The official AA specimen papers instruct candidates to give numerical answers exactly or correct to three significant figures unless the question states otherwise. Avoid rounding intermediate values because this can change the final answer.
Common traps that cost marks
Forgetting to rearrange an equation into the form ax2+bx+c=0
Losing the negative sign in −b when using the quadratic formula
Writing only one solution after taking a square root
Confusing a root x=h with an intercept (h,0)
Giving the vertex when the question asks only for the maximum or minimum value
Assuming roots alone determine the leading coefficient
Including endpoints in a strict inequality
Reporting every calculator root without checking the model’s domain
Using rounded values too early
Sketching a parabola without labelling intercepts, vertex, or relevant scale
After solving, substitute roots back into the original equation when time permits. For a sketch, check whether the direction of opening, intercepts, axis of symmetry, and vertex agree with one another.
How to practise quadratics efficiently
Re-reading notes can refresh definitions, but it does not train method selection. The faster route is an active practice loop:
Attempt one question without looking at the solution.
Mark the exact line where your reasoning failed or became inefficient.
Watch the question worked through step by step.
Close the solution and reproduce the method from memory.
Worked video solutions are most effective when watched after a genuine attempt. Pause before each major step, predict what should happen next, and compare the solution’s structure with your own. Jojo AI can then help explain why an algebraic step is invalid or why a final answer does not satisfy the question.
Conclusion
IB Maths AA quadratics questions become manageable when you recognize their recurring structures. Choose the form that reveals the required information, use the discriminant condition precisely, show the essential reasoning, and check endpoints, signs, domains, and answer format.
For focused revision, combine RevisionDojo’s quadratic notes with targeted Questionbank attempts and worked function videos. Attempting each problem first and then studying its per-question worked solution will build reliable exam technique more quickly than repeatedly reading the same summary.
Emma holds an MMath from the University of Oxford and has taught IB Mathematics for over 20 years, including every year since Analysis & Approaches replaced the old Higher and Standard Level syllabus in 2019. Her focus is IB Mathematics: Analysis & Approaches at SL and HL, developing genuine mathematical intuition from foundational algebra through to the toughest HL topics.
