Deriving the Quadratic Formula

Overview

A step‑by‑step visual derivation of the quadratic formula using completing the square. Viewers see each algebraic manipulation highlighted, with key concepts such as dividing by $a$ and the discriminant emphasized.

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Phases

#	Phase Name	Duration	Description
1	Title & Intro	~4 s	Title "Deriving the Quadratic Formula" fades in, subtitle "Completing the Square" appears briefly, then fades out.
2	Standard Form	~2 s	Write the original quadratic $ax^2+bx+c=0$.
3	Divide by $a$	~3 s	Shift the first line upward, write the divided equation $x^2+\frac{b}{a}x+\frac{c}{a}=0$ with a yellow note "divide by a" that fades in.
4	Move Constant	~3 s	Add the next line $x^2+\frac{b}{a}x = -\frac{c}{a}$ with note "subtract $c/a$"; previous note fades out.
5	Complete the Square – Add Term	~4 s	Move earlier lines upward to make room, write $x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2-\frac{c}{a}$ and show note "add $(b/2a)^2$".
6	Factor Perfect Square	~3 s	Write $\left(x+\frac{b}{2a}\right)^2 = \frac{b^2-4ac}{4a^2}$ with note "factor left side"; previous note fades out.
7	Highlight Discriminant	~3 s	A gold brace appears under the term $b^2-4ac$ with the label "discriminant"; brace and label fade out after a pause.
8	Square Root	~3 s	Write $x+\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac}}{2a}$ with note "take $\pm$ square root"; note fades out.
9	Final Formula	~6 s	All previous lines fade out, the final formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ appears centered, surrounded by a yellow rectangle, title "The Quadratic Formula" appears above, and a callout underneath reads "$b^2-4ac$ is the discriminant" in gold. The discriminant part of the formula is highlighted with an Indicate effect.
10	Outro	~2 s	Everything fades out, leaving a brief pause.

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Layout

┌─────────────────────────────────────────────┐
│                TITLE AREA (TOP)             │
├───────────────────────┬─────────────────────┤
│                       │                     │
│        MAIN AREA      │   (optional) NOTES  │
│   (derivation steps) │   (rarely used)     │
│                       │                     │
├───────────────────────┴─────────────────────┤
│                BOTTOM AREA (CAPTION)        │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Title	Main title and subtitle (phase 1)	Fade in/out with the intro; stays hidden after phase 1
Main	Sequential equations and accompanying yellow notes; later the final formula with surrounding rectangle and discriminant callout	Central focus; equations are stacked vertically, shifting upward as new steps appear
Notes	Optional side notes (not used in this spec)	Reserved for future extensions
Bottom	Small caption for the discriminant callout (appears in phase 9)	Appears only with the final formula

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Notes

• All text (titles, notes, callouts) uses a clean sans‑serif font; titles are white, subtitles and step notes are yellow, discriminant highlights are gold.
• Background color is a dark navy (hex #1a1a2e) throughout.
• Transitions are kept short (0.8–1.5 s) to maintain a brisk pace; total runtime is roughly 30 seconds.
• No additional scenes are required; the entire animation fits within a single Scene class.
• Visual emphasis (SurroundingRectangle, Brace, Indicate) is used instead of extra explanatory text.