IP AMaths Notes (Upper Sec, Year 3-4): 05) Quadratic Equations and Functions

Study guide08 Nov 2025, 00:00 ZUpdated 30 Nov 2025

Discriminant logic, completing the square, and graph interpretation for quadratic functions in IP AMaths.

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Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 05) Quadratic Equations and Functions cover?
A: Discriminant logic, completing the square, and graph interpretation for quadratic functions in IP AMaths.

Quadratics appear everywhere: simultaneous equations, optimisation, and curve sketching. Keep vertex form and discriminant analysis at your fingertips.

Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.

New to the Integrated Programme? Start with What is IP? | Browse all free IP notes.

The core idea is simple: Quadratics connect roots, turning points, and graph shape.

Use it as a working check: Choose the form that answers the question fastest: factorised form for roots, completed-square form for vertex, discriminant for number of real roots.

Then go one layer deeper: Example: to find a minimum value, complete the square first. The constant outside the square is the minimum when the squared part is zero.

Choosing the fastest quadratic form

Before solving or sketching, decide which feature the question is really asking for. Converting to the right form early prevents unnecessary expansion.

Question cue	Best form or test	Why it helps
Find roots, intercepts, or sign intervals	Factored form $y = a(x - \alpha)(x - \beta)$

Reviewed by

Marcus Pang·Managing Director (Maths)

Sources

SEAB GCE O-Level Additional Mathematics (4049) syllabus (examinations from 2025)

Feature to mark	How to get it	What it tells you	Common trap
Opening direction	Check the sign of $a$ .	$a > 0$ opens upward; $a < 0$ opens downward.	Drawing the correct roots but the wrong way up.
Axis and vertex	Use completed-square form or $x = -\frac{b}{2a}$ .	The vertex gives the maximum or minimum point.	Forgetting that the axis is a vertical line, not a point.
Intercepts	Set $x = 0$ for the $y$ -intercept; set $y = 0$ for roots.	Intercepts anchor where the curve crosses the axes.	Treating a repeated root as a crossing instead of a tangent touch.
Sign interval	Read above or below the $x$ -axis after roots are marked.	This solves inequalities such as $y > 0$ or $y < 0$ .	Giving the root values but not the interval between or outside them.

IP AMaths Notes (Upper Sec, Year 3-4): 05) Quadratic Equations and Functions

Choosing the fastest quadratic form

Sources

1 Forms to remember

Quadratic sketching checkpoint

2 Worked example - Completing the square

3 Worked example - Discriminant constraint

4 Worked example - Parameter conditions for positive roots

5 Practice Quiz

6 Try this

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