Unit Overview

Quadratic Functions

CIE IGCSE Additional Mathematics· 5 min read 📊 A high-frequency topic that appears on both Paper 1 and Paper 2

1. Unit at a glance

The unit follows a logical sequence that connects algebraic manipulation to graphical interpretation of quadratics, a key cross-cutting skill for the rest of the syllabus. You will first master completing the square, a foundational technique that lets you quickly find the maximum or minimum output of any quadratic, as well as its full range for a given domain.

You will then move to the discriminant rule, which lets you classify the number and type of roots a quadratic equation has without solving it fully, before combining your algebra and graph knowledge to solve quadratic inequalities and state valid solution ranges.

Work through the following sub-topics in order to build mastery:

Completing the Square, Max/Min and Range

Learn to rewrite quadratic expressions in completed square form to identify vertex coordinates, maximum/minimum values, and calculate function ranges for specified domains.

★★⏱ 10 min

Discriminant, Roots and Quadratic Inequalities

Use the discriminant to classify root types, solve problems involving root conditions, and solve one- and two-sided quadratic inequalities using algebraic and graphical methods.

★★★⏱ 12 min

2. Common Pitfalls

Wrong move:

Forgetting to flip the inequality sign when multiplying/dividing both sides of a quadratic inequality by a negative number

Why:

This reverses the direction of valid solutions, leading to incorrect answer ranges that lose marks in exams

Correct move:

Always reverse the inequality symbol if you multiply or divide across the inequality by a negative value, or use graphical interpretation to verify your solution range

Wrong move:

Calculating the discriminant as $b^{2} + 4 a c$ instead of $b^{2} - 4 a c$

Why:

The sign of the discriminant determines root type, so this sign error completely changes your classification of roots

Correct move:

Memorise the discriminant formula as $Δ = b^{2} - 4 a c$ , and double-check the sign of the $a c$ term before subtracting

Wrong move:

Assuming the vertex of $a x^{2} + b x + c$ is always a minimum value

Why:

The sign of coefficient $a$ determines if the parabola opens upwards (minimum vertex) or downwards (maximum vertex)

Correct move:

Check the $x^{2}$ coefficient first: positive $a$ = minimum vertex, negative $a$ = maximum vertex

3. Quick Reference Cheatsheet

Concept	Formula/Rule	Sub-topic Reference
Completed Square Form	$a (x + h)^{2} + k$	Completing the Square, Max/Min and Range
Vertex Coordinates	$(- h, k)$ for $a (x + h)^{2} + k$	Completing the Square, Max/Min and Range
Discriminant	$Δ = b^{2} - 4 a c$	Discriminant, Roots and Quadratic Inequalities
Equal Roots Condition	$Δ = 0$	Discriminant, Roots and Quadratic Inequalities
Real Distinct Roots Condition	$Δ > 0$	Discriminant, Roots and Quadratic Inequalities
No Real Roots Condition	$Δ < 0$	Discriminant, Roots and Quadratic Inequalities

What's Next

Start your study of this unit with the first sub-topic on completing the square, where you will learn how to rearrange quadratic expressions to quickly find their turning points and range, a skill frequently tested in both short and long exam questions. Once you have mastered all content in this Quadratic Functions unit, you will move on to the next unit covering Factors of Polynomials, which builds directly on the algebraic manipulation skills you practice here.