Deriving the Quadratic Formula

Description

This animation walks through the algebraic derivation of the quadratic formula by completing the square step-by-step. Each line of algebra appears via a MathTex Write or TransformMatchingTex animation, building from ax² + bx + c = 0 all the way to x = (-b ± √(b²-4ac)) / 2a. The discriminant is highlighted in a distinct color to emphasize its role.

---

Phases

#	Phase Name	Duration	Description
1	Setup	4s	Title appears; standard form ax² + bx + c = 0 shown
2	Divide by a	5s	Divide all terms by a; show x² + (b/a)x + c/a = 0
3	Move constant	4s	Subtract c/a: x² + (b/a)x = -c/a
4	Complete the square	8s	Add (b/2a)² to both sides; factor left as (x + b/2a)²
5	Simplify right side	6s	Combine fractions: (b²-4ac)/4a²; highlight discriminant
6	Square root	5s	Take square root of both sides
7	Solve for x	5s	Subtract b/2a; arrive at quadratic formula
8	Final display	7s	Full formula boxed; discriminant color callout

---

Layout

+-----------------------------------------------+
|  Deriving the Quadratic Formula  [title top]  |
|                                                |
|  ax² + bx + c = 0          [step 1]           |
|  x² + (b/a)x + c/a = 0     [step 2]           |
|  x² + (b/a)x = -c/a        [step 3]           |
|  (x + b/2a)² = (b²-4ac)/4a²  [step 4-5]      |
|  x = (-b ± √(b²-4ac)) / 2a  [final]           |
|                                                |
|  discriminant = b²-4ac  [callout, PINK]        |
+-----------------------------------------------+

Area Descriptions

Area	Content	Notes
Top	Title text	Fades after phase 1
Center	Sequential MathTex steps	Each step fades in below the previous
Bottom	Discriminant callout box	Surrounds b²-4ac in PINK/GOLD

---

Assets & Dependencies

• Fonts: LaTeX
• Colors: STEP_COLOR = WHITE, DISCRIMINANT_COLOR = GOLD, HIGHLIGHT = PINK, BG = #1a1a2e
• Manim version: ManimCE 0.19.1

---

Notes

• Steps are shown sequentially; earlier steps shift upward as new ones appear
• SurroundingRectangle used to box the discriminant
• Final formula gets a large SurroundingRectangle in YELLOW