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Completing the Square for Quadratic Equation (II) : Solving the Equation

Completing the Square for Quadratic Equation (II) : Solving the Equation
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2017-03-04 23:57
FunctionGeometric ConstructionSlider Animation
This example demonstrates how to solve quadratic equations using the method of completing the square, involving core mathematical concepts such as completing the square and solving quadratic equations. Construction highlights extracted from the 'Construction Summary' using GeoGebra commands/objects: Point, Midpoint, RandomElement, Vector, Random, Simplify, Intersect, Polygon, Dilate, Translate, Root; objects: n_{motion}, P_{xsquared}, h, D, B, M_{BD}, a, u_1, c, b, sl1, a2, b2, sl2, sr2, s3, s4, as, bs, sl5, sr5, sl6, d, f, button1, j, m, k, l, r, e, textInstructionSlider1, textFinalText, p, q, g, i, u, t, s. Suitable for high school mathematics teaching, helping students understand and master the steps and methods of solving quadratic equations using the completing square method.

GeoGebra Commands & Parameters

  1. Point(yAxis) → D
  2. Point(xAxis) → B
  3. Midpoint(B, D) → M_{BD}
  4. RandomElement({-7, -6, -5, -4, -3, -2, -1, 7, 6, 5, 4, 3, 2, 1}) → a
  5. Vector(P_{xsquared}, M_{BD}) → u_1
  6. Random(-7, 7) → b
  7. Simplify((x - a)^(2) - c^(2)) → sl1
  8. Simplify(y^(2) + (b2 * y)) → sr2
  9. Simplify((-((x - a)^(2) - c^(2) - x^(2) - (a2 * x)))) → s3
  10. Simplify(x^(2) + (a2 * x)) → s4
  11. Simplify((x - a)^(2)) → sl5
  12. Simplify((y - b)^(2)) → sr5
  13. Simplify(x - h) → sl6
  14. Intersect(xAxis, yAxis) → A
  15. Polygon(A, B, C, D) → poly1, a_1, b_1, c_1, d_1
  16. Point(xAxis) → E
  17. Point(xAxis) → F
  18. Polygon(E, F, G, H) → poly2, e_1, f_1, g_1, h_1
  19. Polygon(I, J, K, L) → poly3, i_1, j_1, k_1, l_1
  20. Polygon(M, N, O, P) → poly4, m_1, n_1, o_1, p_1
  21. Dilate(M, h / a, O) → M'
  22. Dilate(N, h / a, O) → N'
  23. Dilate(O, h / a, O) → O'
  24. Dilate(P, h / a, O) → P'
  25. Polygon(M', N', O', P') → poly4', m', n', o', p'
  26. Midpoint(A, B) → M_{AB}
  27. Midpoint(A, D) → M_{AD}
  28. Midpoint(E, F) → M_{EF}
  29. Midpoint(F, H) → M_{FH}
  30. Midpoint(I, L) → M_{IL}
  31. Midpoint(J, L) → M_{JL}
  32. Midpoint(N', P') → M_{NP}
  33. Translate(P_{xsquared}, Vector[(n_{motion} * u_1)]) → R
  34. Vector(P_{xsquared}, R) → v_1
  35. Vector(P_{bx}, M_{FH}) → u_2
  36. Translate(P_{bx}, Vector[(n_{motion} * u_2)]) → Q
  37. Vector(P_{bx}, M_{JL}) → u_3
  38. Translate(P_{bx}, Vector[(n_{motion} * u_3)]) → S
  39. Vector(P_{bx}, Q) → v_2
  40. Vector(P_{bx}, S) → v_3
  41. Dilate(M, j / a^(2), O) → M'_1
  42. Dilate(N, j / a^(2), O) → N'_1
  43. Dilate(O, j / a^(2), O) → O'_1
  44. Dilate(P, j / a^(2), O) → P'_1
  45. Polygon(M'_1, N'_1, O'_1, P'_1) → poly4'_1, m'_1, n'_1, o'_1, p'_1
  46. Random((-(a2 / 2)^(2)) + 1, 8) → c_R
  47. Root(x^(2) + (a2 * x) - c_R) → T_{1st of roots}, U_{2nd of roots}

AI Drawing Prompt Reference

Draw a quadratic equation graph and solve it using the completing the square method. Start by creating points D and B on the coordinate axes, then randomly generate values for a and b. Construct the quadratic equation graph with translations and scaling transformations, and finally obtain the two roots T_{1st of roots} and U_{2nd of roots}.
The prompt above was compiled from the original drawing commands and is for reference only.