Factoring a Quadratic and Visualizing Its Roots

Scene: QuadraticSolutionScene (single scene)

Purpose: Visually demonstrate solving the quadratic equation $x^2 - 5x + 6 = 0$ by factoring, confirming roots with a graph, and highlighting the solution set {2,3}.

Duration: ~18 seconds.

1. Introduction (0‑2 s)

• Fade‑in a centered MathTex displaying the equation $$x^2 - 5x + 6 = 0$$ in white on a dark background.
• Background rectangle with slight opacity behind the text for readability.

2. Factoring Step (2‑7 s)

• Transform the equation into a factorized form using a smooth Transform:
  $$x^2 - 5x + 6 = (x-2)(x-3) = 0$$
• Highlight each factor sequentially:
  • First factor $(x-2)$ turns teal for 0.5 s.
  • Second factor $(x-3)$ turns orange for 0.5 s.
• Add a brief ‘zero‑product property’ note as a small MathTex box: ‘If $AB=0$ then $A=0$ or $B=0$’, with an opaque light‑gray background.

3. Solving for Roots (7‑10 s)

• Split the equation into two separate lines:
  $$\begin{cases} x-2 = 0 \\ x-3 = 0 \end{cases}$$
• Animate each line appearing with a Write effect.
• Immediately transform each line into the solutions $x = 2$ and $x = 3$ using Transform.
• Collect the solutions into a set displayed as {2,,3} with a brief pulse.

4. Graphical Confirmation (10‑16 s)

• Introduce a coordinate plane (axes) sliding in from the left.
• Plot the parabola $y = x^2 - 5x + 6$ in a smooth Create animation, using a deep blue curve.
• Mark the x‑intercepts at $x=2$ and $x=3$ with red dots and small vertical lines down to the axis.
• When each dot appears, display the corresponding solution number (2 or 3) next to it, each inside a small opaque rectangle.
• Zoom the camera slightly toward the intercepts during the last 2 s to emphasize the roots.

5. Summary (16‑18 s)

• Fade‑out the graph while keeping the solution set {2,3} centered.
• Add a final ‘Solution’ label above the set, with a contrasting background, then hold for 1 s before fading out everything.

Visual Style

• Colors: Equation text – white; factors – teal & orange; parabola – deep blue; intercept dots – red; solution set – green.
• Backgrounds for text: Light‑gray rectangles with 0.8 opacity.
• Transitions: Use FadeIn, Transform, Write, Create, and gentle Zoom for emphasis.
• Camera: Start with a static view; during the graph stage, perform a smooth camera.frame zoom toward the intercepts.

Timing Summary

Segment	Duration
Introduction	2 s
Factoring	5 s
Solving roots	3 s
Graphical confirmation	6 s
Summary	2 s
Total	≈18 s

All elements fit within a single Manim Scene class. No additional text beyond the mathematical expressions is required.