Quadratics & Polynomials (AA HL)
Quadratics are the simplest curved functions you'll meet, and they set up the ideas used for every higher-degree polynomial afterwards: roots, factors, and how a function's coefficients encode information about its graph without you having to solve anything. This topic covers the discriminant, the quadratic formula, the factor theorem, and the sum and product of a polynomial's roots - including polynomials with complex roots.
What the syllabus says
This topic maps onto three points in the official IB Analysis & Approaches syllabus, two at SL and one AHL extension.
| Code | Syllabus content |
|---|---|
| SL2.6 | The quadratic function \(f(x)=ax^2+bx+c\): its graph, \(y\)-intercept \((0,c)\), axis of symmetry. The forms \(f(x)=a(x-p)(x-q)\) with \(x\)-intercepts \((p,0)\) and \((q,0)\), and \(f(x)=a(x-h)^2+k\) with vertex \((h,k)\). Students are expected to convert between forms. |
| SL2.7 | Solving quadratic equations and inequalities by factorising, completing the square, and the quadratic formula. The discriminant \(\Delta=b^2-4ac\) and the nature of the roots - two distinct real roots, two equal real roots, or no real roots. |
| AHL2.12 | Polynomial functions, their graphs and equations; zeros, roots and factors. The factor and remainder theorems. Sum and product of the roots of a polynomial equation, read directly from its coefficients. |
The quadratic content is core SL, examinable at HL; the general polynomial-root results are AHL-only extensions.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is the discriminant?
The discriminant \(\Delta=b^2-4ac\) is a single number, calculated from a quadratic's coefficients, that tells you how many real roots the equation has without solving it. Its sign - positive, zero, or negative - sorts every quadratic into one of three types.
e.g. For \(x^2-6x+9=0\): \(\Delta=36-36=0\), so the equation has one repeated root, \(x=3\).
What are the roots of a quadratic?
The roots (or zeros) of a quadratic are the values of \(x\) that make \(f(x)=0\) - where the graph crosses or touches the \(x\)-axis. They can be found by factorising, completing the square, or the quadratic formula.
e.g. \(x^2-5x+6=0\) factorises as \((x-2)(x-3)=0\), so the roots are \(x=2\) and \(x=3\).
What are the sum and product of roots?
For \(ax^2+bx+c=0\) with roots \(\alpha\) and \(\beta\), the sum \(\alpha+\beta=-\tfrac{b}{a}\) and the product \(\alpha\beta=\tfrac{c}{a}\) - you can read these directly from the coefficients without ever solving the equation.
e.g. \(x^2-7x+12=0\) has roots 3 and 4: sum \(=7=-\tfrac{-7}{1}\), product \(=12=\tfrac{12}{1}\).
What is a repeated root?
A repeated (or double) root happens when a factor appears more than once in a polynomial, so the graph touches the \(x\)-axis at that point instead of crossing it. For a quadratic this happens exactly when \(\Delta=0\).
e.g. \(x^2-4x+4=(x-2)^2=0\) has the repeated root \(x=2\).
What is the factor theorem?
The factor theorem says that if \(p(a)=0\) for a polynomial \(p(x)\), then \((x-a)\) is a factor of \(p(x)\) - and conversely. It's the standard way to test a possible root before dividing a polynomial.
e.g. For \(p(x)=x^3-6x^2+11x-6\): \(p(1)=1-6+11-6=0\), so \((x-1)\) is a factor.
Key formulas
Two formulas handle every quadratic on this topic, and one general result extends them to any polynomial. The two tables below summarise them at a glance - the explanations underneath go into more depth on each one.
Formula reference
The quadratic formula, the discriminant, and the general sum/product-of-roots result are all given in the formula booklet.
| Formula | Used for | Booklet? |
|---|---|---|
| \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\) | Quadratic formula | ✓ Yes |
| \(\Delta=b^2-4ac\) | Discriminant - nature of roots | ✓ Yes |
| \(\sum \alpha_i=-\dfrac{a_{n-1}}{a_n},\quad \prod \alpha_i=(-1)^n\dfrac{a_0}{a_n}\) | Sum and product of roots (any degree) | ✓ Yes |
| \(h=-\dfrac{b}{2a}\) | Axis of symmetry / vertex \(x\)-coordinate | Not in booklet |
| If \(p(a)=0\) then \((x-a)\mid p(x)\) | Factor theorem | Not in booklet |
Quadratic roots vs cubic roots
The sum-and-product idea extends naturally from a quadratic to a cubic, just with an extra root to track.
| Feature | Quadratic \(ax^2+bx+c\) | Cubic \(ax^3+bx^2+cx+d\) |
|---|---|---|
| Roots | \(\alpha,\beta\) | \(\alpha,\beta,\gamma\) |
| Sum of roots | \(\alpha+\beta=-\tfrac ba\) | \(\alpha+\beta+\gamma=-\tfrac ba\) |
| Product of roots | \(\alpha\beta=\tfrac ca\) | \(\alpha\beta\gamma=-\tfrac da\) |
| Complex roots (real coefficients) | Occur as one conjugate pair, or none | Occur in pairs, so at least one root is real |
Solving quadratics
All three methods below always work - the fastest choice depends on how the quadratic is presented.
Factorising
\[ax^2+bx+c=a(x-p)(x-q)\]
Quickest when the roots are rational and easy to spot by inspection.
Completing the square
\[a(x-h)^2+k,\quad h=-\tfrac{b}{2a}\]
Gives the vertex directly - the natural method for finding a maximum or minimum.
Quadratic formula
\[x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\]
Always works, including for irrational or complex roots.
The discriminant
The sign of \(\Delta=b^2-4ac\) sorts every quadratic into exactly one of these three cases.
\(\Delta>0\)
Two distinct real roots - the graph crosses the \(x\)-axis at two points.
\(\Delta=0\)
One repeated real root - the graph touches the \(x\)-axis at its vertex.
\(\Delta<0\)
No real roots - the graph doesn't meet the \(x\)-axis; the roots form a complex conjugate pair.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Consider \(x^2-6x+c=0\).
(a) Write down the discriminant in terms of \(c\).
(b) Find the value of \(c\) for which the equation has equal roots.
(c) State the repeated root.
Worked solution
(a) \(\Delta=36-4c.\) A1
(b) Equal roots \(\Rightarrow36-4c=0\Rightarrow c\) M1
\(=9.\) A1
(c) \(x=\dfrac{6}{2}=3.\) A1
The roots of \(2x^2 - 6x + 5 = 0\) are \(\alpha\) and \(\beta\).
(a) Find \(\alpha+\beta\).
(b) Find \(\alpha\beta\).
(c) Find \(\alpha^2+\beta^2\).
Worked solution
(a) \(\alpha + \beta = -\dfrac{-6}{2} = 3.\) M1
(b) \(\alpha\beta = \dfrac{5}{2}.\) A1
(c) \(\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = 9 - 5\) M1
\(= 4.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Forgetting the \(\pm\) when taking a square root. Whether it's the quadratic formula or solving \(a(x-h)^2=k\) directly, a square root always has two values unless it's exactly zero.
- Writing the sum of roots as \(-b\) instead of \(-\tfrac ba\). This only looks the same when \(a=1\) - forgetting to divide by the leading coefficient is one of the most common slips in Vieta's formulas.
- Assuming complex roots always come in conjugate pairs. That's only guaranteed when every coefficient of the polynomial is real - it doesn't apply to a polynomial with complex coefficients.
- Confusing \(\Delta=0\) with \(\Delta<0\). A zero discriminant means one repeated real root, not "no roots" - only a negative discriminant means no real roots at all.
Using your GDC
There isn't a single fixed button sequence for this topic, since it spans both no-calculator algebra and calculator-assisted graphing.
On Paper 1, questions on the discriminant, the factor theorem, and sum/product of roots are done entirely without a calculator - a GDC won't help with that algebra. On Paper 2, though, your calculator is genuinely useful: graphing a polynomial and using its root-finding (zero) tool is the fastest way to locate real roots or confirm a factorisation, and most calculators can also solve a polynomial equation directly using an equation solver or polynomial-root finder. Always double-check that a root a calculator gives you is exact where the question demands an exact answer, since graphical tools usually only return a rounded decimal.
See the full GDC guide for calculator-specific graphing and equation-solving steps.
Ready to practise properly?
Quadratics & polynomials questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What does the discriminant tell you?
The discriminant \(\Delta=b^2-4ac\) tells you how many real roots a quadratic has, without solving it. \(\Delta>0\) gives two distinct real roots, \(\Delta=0\) gives one repeated real root, and \(\Delta<0\) gives no real roots (a complex conjugate pair instead).
How do I find the sum and product of a polynomial's roots without solving it?
Read them straight from the coefficients. For a quadratic \(ax^2+bx+c\), the sum of roots is \(-b/a\) and the product is \(c/a\). Higher-degree polynomials follow the same pattern with alternating signs, using the leading and constant coefficients.
Do complex roots of a polynomial always come in pairs?
Yes, but only when every coefficient of the polynomial is real. In that case, if \(a+bi\) is a root, its conjugate \(a-bi\) must also be a root.
Can I use my GDC to find roots and factorise polynomials?
Yes on Paper 2 - graphing and root-finding on a GDC is a fast way to locate real zeros or check a factorisation. Paper 1 questions on the discriminant, the factor theorem, and sum/product of roots are done without a calculator, so the algebra still needs to be secure. See the GDC guide for model-specific instructions.
Sub-topics
Quadratics & Polynomials broken down into its individual skills, each with its own focused page.
Related topics
More Functions topics from the same AA HL syllabus unit, in case you want to keep going.