Solving a Quadratic by Factorisation

Overview

A concise step‑by‑step visual proof of solving the quadratic equation $x^{2}+5x+6=0$ using factorisation. The animation demonstrates each algebraic transformation, highlighting the reasoning process so students see how the solution emerges.

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Phases

#	Phase Name	Duration	Description
1	Scene 1 – The Problem	~4 s	The equation $x^{2}+5x+6=0$ appears term by term, then the full equation is shown.
2	Scene 2 – Identify Method	~3 s	The quadratic part is highlighted and the label “METHOD: FACTORISATION” appears.
3	Scene 3 – Find Two Numbers	~5 s	The product and sum conditions are introduced, then the numbers 2 and 3 are revealed and linked to both conditions.
4	Scene 4 – Split Middle Term	~4 s	The term $5x$ splits into $2x+3x$ and the equation becomes $x^{2}+2x+3x+6=0$.
5	Scene 5 – Group & Factor	~6 s	Brackets group the terms, common factors are taken, the repeated factor $(x+2)$ is highlighted, and the equation becomes $(x+2)(x+3)=0$.
6	Scene 6 – Zero‑Product Property	~3 s	The product equation splits into two branches $x+2=0$ and $x+3=0$.
7	Scene 7 – Solve for $x$	~5 s	Each branch is solved stepwise, yielding $x=-2$ and $x=-3$; both solutions are displayed together.
8	Scene 8 – Verify & Conclude	~5 s	Substitutions $x=-2$ and $x=-3$ are shown to satisfy the original equation, then the final conclusion $\therefore x=-2$ or $x=-3$ is held on screen for 3 s.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│                MAIN (primary visual)        │
│                                             │
├─────────────────────────────────────────────┤
│  Caption / label (optional, small text)     │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	All equations, brackets, arrows, highlights, and grouping visuals.	Center‑aligned, large white typography on a dark neutral background.
Caption	Short persistent labels such as “METHOD: FACTORISATION”.	Appears only in scenes that require a textual cue; otherwise hidden.

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Notes

• Assumptions: total runtime ~35 s, each scene paced for a class‑8/9/10 audience; smooth linear fades and zooms are used but not detailed in code.
• Camera zooms subtly when focusing on a newly highlighted term (e.g., zoom into $5x$ before splitting).
• All highlights use a bright cyan outline; arrows are thin white lines with arrowheads.
• Brackets animate from the sides to enclose grouped terms.
• The final answer remains on screen for at least 3 seconds as required.
• No additional graphics, characters, or narration text are included in the visual layers; narration is assumed to be provided separately.
• The specification is designed for a single Manim Scene class.
• All mathematical symbols are rendered exactly as given, preserving superscripts and minus signs.