Deriving the Quadratic Formula
Audience: High SchoolCategory: Math
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 axΒ² + bx + c = 0 all the way to x = (-b Β± β(bΒ²-4ac)) / 2a. The discriminant is highlighted in a distinct color to emphasize its role.
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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 axΒ² + bx + c = 0 all the way to x = (-b Β± β(bΒ²-4ac)) / 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 axΒ² + bx + c = 0 shown |
| 2 | Divide by a | 5s | Divide all terms by a; show xΒ² + (b/a)x + c/a = 0 |
| 3 | Move constant | 4s | Subtract c/a: xΒ² + (b/a)x = -c/a |
| 4 | Complete the square | 8s | Add (b/2a)Β² to both sides; factor left as (x + b/2a)Β² |
| 5 | Simplify right side | 6s | Combine fractions: (bΒ²-4ac)/4aΒ²; 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] |
| |
| axΒ² + bx + c = 0 [step 1] |
| xΒ² + (b/a)x + c/a = 0 [step 2] |
| xΒ² + (b/a)x = -c/a [step 3] |
| (x + b/2a)Β² = (bΒ²-4ac)/4aΒ² [step 4-5] |
| x = (-b Β± β(bΒ²-4ac)) / 2a [final] |
| |
| discriminant = bΒ²-4ac [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 bΒ²-4ac 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