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Deriving the Quadratic Formula via Completing the Square

Deriving the Quadratic Formula via Completing the Square
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2018-02-04 19:52
FunctionCoordinate GeometryDerivativeOptimizationSlider Animation
This example demonstrates how to derive the quadratic formula using the method of completing the square through an animation. Key math concepts involved include quadratic equations, completing the square, and algebraic manipulation. The construction utilizes GeoGebra commands and objects such as Polygon, Ray, Point, OrthogonalLine, Line, Intersect, If, Rotate, Midpoint, and Vector. Suitable for high school mathematics education to help students understand the origin and derivation of the quadratic formula.

GeoGebra Commands & Parameters

  1. Polygon(A_1, A_2, 4) → poly1, b_1, b, c, d, A_3, A_4
  2. Ray(A_2, A_2 + Vector[A_1, A_2]) → r
  3. Point(r) → B_1
  4. OrthogonalLine(B_1, b_1) → e
  5. Line(A_4, A_3) → f
  6. Intersect(e, f) → B_4
  7. If(a < 25, 0, If[a ≤ 125, (pi * (a - 25) / 100)]) → a_1
  8. OrthogonalLine(B_2, b_1) → g
  9. Intersect(f, g) → B_3
  10. Rotate(B_3, a_1, B_4) → B_3'
  11. Rotate(B_1, a_1, B_4) → B_1'
  12. Rotate(B_2, a_1, B_4) → B_2'
  13. Polygon(B_4, B_1', B_2', B_3') → poly2, b_4, b_1', b_2', b_3'
  14. If(a ≥ 175, 1.5707963267948966, If[125 < a, (pi / 2 * (a - 125) / 50)]) → a_2
  15. Rotate(B_4, a_2, A_3) → B_4'
  16. Rotate(C_1, a_2, A_3) → C_1'
  17. Rotate(C_2, a_2, A_3) → C_2'
  18. Polygon(A_3, B_4', C_1', C_2') → poly3, a_3, b_4', c_1', c_2'
  19. Rotate(B_3, 90°, B_4) → D
  20. If(a ≥ 175, Polygon[A_3, B_4, D, B_4']) → poly4
  21. Polygon(A_2, B_1, B_4, A_3) → poly5, a_4, b_2, b_3, a_5
  22. Polygon(T_1, T_2, T_3, T_4) → polyTbox, a_6, b_5, c_1, e_1
  23. Midpoint(b_3) → A
  24. Midpoint(a_4) → B
  25. Midpoint(b_1) → C
  26. Midpoint(d) → E
  27. Midpoint(c_1') → F
  28. Midpoint(b_1') → G
  29. Vector(H, I) → u

AI Drawing Prompt Reference

Create an animated geometric construction showing the derivation of the quadratic formula. Start with a square, extend a ray from one vertex, place a point on it, and draw a perpendicular line intersecting a diagonal to form a new quadrilateral. Use a slider to rotate parts of the figure around vertices, gradually rearranging shapes into rectangles and squares to illustrate area relationships. Include multiple polygons, midpoints, and a vector to highlight the algebraic manipulation.
The prompt above was compiled from the original drawing commands and is for reference only.