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

Study guide08 Nov 2025, 00:00 ZUpdated 17 Jul 2026

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

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For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.

How this chapter applies

Eclat core: completed-square extrema, sign conditions, quadratic models, discriminant conditions, line-curve intersections, and quadratic inequalities form the main route.
School-sensitive extension: parameter-heavy optimisation and unfamiliar modelling contexts may vary by school.
2027 national comparison: K341 Topics A1 and A2 cover extrema, always-positive or always-negative conditions, models, root and intersection conditions, and quadratic inequalities.
Check your school: confirm the expected proof language for discriminant and sign conditions.
Exam-track route: use the separate O-Level and SEC G3 Additional Mathematics notes for K341 topic ownership.

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.

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Marcus Pang·Managing Director (Maths)

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

How this chapter applies

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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