Consider the quadratic equation $x^2 + (k+2)x + 2k = 0$ , where $k \in \mathbb{R}$ . Show algebraically that this equation will always have real roots, regardless of the value of $k$ .

To determine the nature of the roots, calculate the discriminant $\Delta = b^2 - 4ac$ . (M1) Here, $a = 1$ , $b = (k+2)$ , and $c = 2k$ . $\Delta = (k+2)^2 - 4(1)(2k)$ . (M1) Expand and simplify: $\Delta = k^2 + 4k + 4 - 8k$ . (A1) $\Delta = k^2 - 4k + 4$ . (A1) Factorise: $\Delta = (k - 2)^2$ . Since any real number squared is non-negative, $(k - 2)^2 \ge 0$ for all $k \in \mathbb{R}$ . Therefore, the equation always has real roots. R1

Home › IB Maths AA SL › Questions by topic › Quadratic Functions

IB Maths AA SL · Unit 2: Functions

Quadratic Functions (IB Maths AA SL): notes, exam questions and worked examples

Q: Is the discriminant in the IB formula booklet?

The quadratic formula and the discriminant b² − 4ac are in the AA SL booklet. What each sign of the discriminant means (two, one or no real roots) is not printed, so learn it.

Q: How do I find the vertex of a quadratic?

Complete the square to get a(x − h)² + k; the vertex is (h, k). Or use x = −b / 2a from the booklet and substitute to find the y-coordinate.

Q: How many IB Maths AA SL quadratic functions questions are there?

There are 39 exam-style quadratic functions questions in the AA SL question bank (Paper 1: 31 · Paper 2: 8), graded 8 easy, 19 medium, 8 hard, 4 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is quadratic functions on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 31 · Paper 2: 8. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Updated October 2026 · By Pete Bromfield

A quadratic $y = a x^{2} + b x + c$ has a parabola as its graph, with axis of symmetry $x = - \frac{b}{2 a}$ . Write it in vertex form to read the turning point, factorise or use the formula to find the roots, and use the discriminant $Δ = b^{2} - 4 a c$ to say how many real roots there are without solving.

39 questions
Paper 1: 31
Paper 2: 8
3 worked examples

Practise quadratic functions questions →

Free account: 5 more practice questions on quadratic functions with instant feedback, and full worked answers on the free questions. Free, no card. Join free →

Notes
Formulas
Worked examples
Questions
FAQ

Quadratic Functions notes

Three forms of a quadratic

Standard form $a x^{2} + b x + c$ gives the $y$ -intercept $c$ . Vertex form $a (x - h)^{2} + k$ gives the vertex $(h, k)$ . Intercept form $a (x - p) (x - q)$ gives the roots $p$ and $q$ . Complete the square to move from standard to vertex form: $2 x^{2} - 8 x + 5 = 2 (x - 2)^{2} - 3$ , so the vertex is $(2, - 3)$ .

Solving quadratic equations

Rearrange to $= 0$ first. Factorise when you can; otherwise use $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ . On Paper 2 the GDC finds roots, but show the equation you solved.

The discriminant

$Δ = b^{2} - 4 a c$ . If $Δ > 0$ there are two distinct real roots, if $Δ = 0$ one repeated root (the graph touches the $x$ -axis), and if $Δ < 0$ no real roots. For example $x^{2} + k x + 9 = 0$ has equal roots when $k^{2} - 36 = 0$ , so $k = \pm 6$ .

Where a line meets a curve

Substitute the line into the quadratic and look at the discriminant of the result: two points, one point (the line is a tangent) or none. The line $y = m x$ is a tangent to $y = x^{2} + 3$ when $x^{2} - m x + 3 = 0$ has $Δ = 0$ , so $m^{2} = 12$ and $m = \pm 2 \sqrt{3}$ .

Always positive or always negative

$a x^{2} + b x + c > 0$ for every $x$ exactly when $a > 0$ and $Δ < 0$ : the parabola opens upwards and never meets the $x$ -axis.

Key formulas

Quadratic: axis of symmetry In the booklet: $x = - \frac{b}{2 a}$
Quadratic formula; discriminant In the booklet: $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}, Δ = b^{2} - 4 a c$
Roots: $Δ > 0$ two distinct real, $Δ = 0$ one repeated, $Δ < 0$ no real roots Learn it
Vertex form, vertex $(h, k)$ ; intercept form Learn it: $y = a (x - h)^{2} + k; y = a (x - p) (x - q)$

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AA SL formulas →

Explained, with a worked example: Quadratic formula

Quadratic Functions worked examples

Worked example 1: Vertex and intercepts algebraically · easy

The quadratic $f (x) = - 2 (x - 3)^{2} + 8$ . Find the exact coordinates of the vertex, the $y$ -intercept, and the $x$ -intercepts.

Solution

4 steps

Examiner tip: When taking the square root to find

x

-intercepts, always include the

\pm

symbol or you will lose one of the roots.

Worked example 2: Discriminant for number of roots · medium

The equation $x^{2} + k x + 9 = 0$ has two equal real roots. Find the possible exact values of $k$ .

Solution

4 steps · 1 to fill in

Examiner tip: "Two equal real roots", "touches the

x

-axis", and "is a perfect square" all signal setting the discriminant exactly to zero.

Worked example 3: Tangents and the discriminant · hard

Find the exact values of $m$ for which $y = m x - 4$ is a tangent to $y = x^{2} + 3 x + 5$ .

Solution

4 steps · 1 to fill in

Examiner tip: Always group

x

coefficients in brackets before calculating the discriminant. Expanding prematurely invites messy algebraic errors.

Play with it Roots and y-intercept: drag the sliders

Change a, p and q so that y = a(x − p)(x − q) crosses the x-axis at −1 and 3 and the y-axis at −6. Then expand your answer into the form y = ax² + bx + c.

Want more? Open the full Grapher.

Quadratic Functions exam-style questions

3 questions from our AA SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Each has a full mark scheme in the practice area.

Question 1 · easy · 4 marks · Paper 2

The function $h (x) = - 0.65 x^{2} + 5.2 x + 3.1$ models the trajectory of an object.

Sketch the graph of $y = h (x)$ on your graphic display calculator.
Use the maximum feature of your GDC to find the range of the function $h (x)$ .

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

Let $f (x) = x^{2} - 6 x + 14$ .

By completing the square, express $f (x)$ in the form $(x - h)^{2} + k$ .
Hence, or otherwise, explain algebraically why the graph of $f (x)$ does not intersect the $x$ -axis.

Attempt it and see the mark scheme →

Question 3 · hard · 5 marks · Paper 1

Consider the quadratic equation $x^{2} + (k + 2) x + 2 k = 0$ , where $k \in R$ . Show algebraically that this equation will always have real roots, regardless of the value of $k$ .

Full worked solution Free, no sign-in

To determine the nature of the roots, calculate the discriminant $Δ = b^{2} - 4 a c$ . (M1)
Here, $a = 1$ , $b = (k + 2)$ , and $c = 2 k$ .
$Δ = (k + 2)^{2} - 4 (1) (2 k)$ . (M1)
Expand and simplify: $Δ = k^{2} + 4 k + 4 - 8 k$ . (A1)
$Δ = k^{2} - 4 k + 4$ . (A1)
Factorise: $Δ = (k - 2)^{2}$ .
Since any real number squared is non-negative, $(k - 2)^{2} \geq 0$ for all $k \in R$ . Therefore, the equation always has real roots. R1

Every question here has a full worked solution. See them all →

Free account — sign in to download Topic PDF: 7 questions with worked answers · 4 pages · 195 KB

All 39 quadratic functions questions →

Quadratic Functions on your calculator

A GDC is allowed on Paper 2, not on Paper 1. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Graphs: intersections, zeros, max and min on your calculator

Quadratic Functions FAQ

Is the discriminant in the IB formula booklet?

The quadratic formula and the discriminant b² − 4ac are in the AA SL booklet. What each sign of the discriminant means (two, one or no real roots) is not printed, so learn it.

How do I find the vertex of a quadratic?

Complete the square to get a(x − h)² + k; the vertex is (h, k). Or use x = −b / 2a from the booklet and substitute to find the y-coordinate.

How many IB Maths AA SL quadratic functions questions are there?

There are 39 exam-style quadratic functions questions in the AA SL question bank (Paper 1: 31 · Paper 2: 8), graded 8 easy, 19 medium, 8 hard, 4 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is quadratic functions on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 31 · Paper 2: 8. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Next steps

More AA SL Unit 2: Equations of Straight Lines · Exponential and Logarithmic Functions · Functions: Domain, Range & Inverses · Graphing Functions · Rational and Reciprocal Functions

Teaching this topic? Free ready-to-teach lessons (slides and teacher notes): Quadratic graphs & vertex · Discriminant & roots · Solving quadratics (factor, formula, complete square).

Keep revising AA SL: AA SL revision notes · IB Maths AA SL past papers · AA SL common exam mistakes · AA SL syllabus checklist · AA SL functions knowledge organiser

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

Join free

Already have an account? Sign in

← All IB Maths AA SL topics

Quadratic Functions (IB Maths AA SL): notes, exam questions and worked examples

Try it first

Quadratic Functions notes

Three forms of a quadratic

Solving quadratic equations

The discriminant

Where a line meets a curve

Always positive or always negative

Key formulas

Quadratic Functions worked examples

Worked example 1: Vertex and intercepts algebraically · easy

Solution

Worked example 2: Discriminant for number of roots · medium

Solution

Worked example 3: Tangents and the discriminant · hard

Solution

Quadratic Functions exam-style questions

Question 1 · easy · 4 marks · Paper 2

Question 2 · medium · 5 marks · Paper 1

Question 3 · hard · 5 marks · Paper 1

Full worked solution Free, no sign-in

Quadratic Functions on your calculator

Quadratic Functions FAQ

Next steps

What else is free here