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因式分解法解二次方程

因式分解法解二次方程
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2013-05-05 16:18
函数
这个范例用于演示如何通过因式分解法解二次方程。涉及的核心数学概念包括二次方程、因式分解和根的求解。范例使用了 GeoGebra 命令如 If, GCD, Simplify, Factor, Textfield, Coefficients, FractionText, Intersect 等，以及对象如 positivecoefficientsonly, signchanger, easylevel, listnegative, upperlimitInteger, textroot2i, b_{2r}, b_{1r}, coefficientofxsquaredoneonly, a_{1r}, k_1, a_{2r}, s_{a1}, s_1, s_2, cs_1, k_2, cs_2, a_1, b_2, a_2, b_1, f_2, f_1, textTitle, listCoefficientsExpected, k, f_c, f, Steps, textGraphicshorizontallinedotted, textGraphicsverticallinedotted, a_{i1}, a_{i2}, b_{i1}, b_{i2}, inputBox1, inputBox2, inputBox3, inputBox4。适合初中及以上学生用于课堂演示和自我练习。

GeoGebra 命令及参数

  1. If(positivecoefficientsonly, 1, -1) → signchanger
  2. If(easylevel, 4, 7) → upperlimitInteger
  3. If(coefficientofxsquaredoneonly, 1, (Random[1, upperlimitInteger] * Element[listnegative, Random[1, 3]])) → a_{1r}
  4. GCD(a_{1r}, b_{1r}) → k_1
  5. If(coefficientofxsquaredoneonly, 1, (Random[1, upperlimitInteger] * Element[listnegative, Random[1, 3]])) → a_{2r}
  6. If(a_{1r} < 0 ∧ a_{2r} < 0, -1, 1) → s_{a1}
  7. If(a_{2r} < 0 ∧ b_{2r} < 0 ∧ (a_{1r} * b_{1r}) < 0 ∨ a_{1r} < 0 ∧ b_{1r} < 0, -1, 1) → s_1
  8. If(a_{1r} < 0 ∧ b_{1r} < 0 ∧ (a_{2r} * b_{2r}) < 0 ∨ a_{2r} < 0 ∧ b_{2r} < 0, -1, 1) → s_2
  9. GCD(a_{2r}, b_{2r}) → k_2
  10. If(a_2 < 0 ∧ b_2 > 0, b_2 + (a_2 * x), Simplify[(a_2 * x) + b_2]) → f_2
  11. If(a_1 < 0 ∧ b_1 > 0, b_1 + (a_1 * x), Simplify[(a_1 * x) + b_1]) → f_1
  12. GCD(listCoefficientsExpected) → k
  13. Simplify(Expand[(f_1(x) * f_2(x))]) → f_c
  14. Factor(f_c) → f
  15. Textfield(a_{i1}) → inputBox1
  16. Textfield(a_{i2}) → inputBox2
  17. Textfield(b_{i1}) → inputBox3
  18. Textfield(b_{i2}) → inputBox4
  19. Simplify(Expand[(((a_{i1} * x) + b_{i1}) * ((a_{i2} * x) + b_{i2}))]) → f_i
  20. Simplify((a_{i1} * x) + b_{i1}) → f_{c1}
  21. Simplify((a_{i2} * x) + b_{i2}) → f_{c2}
  22. Coefficients(f_i) → listCoefficientsInputFunction
  23. If(abs(a_1) ≟ 1, If[a_1 > 0, "1 ", "-1"], "" + a_1) → textcoeffa1
  24. If(abs(a_2) ≟ 1, If[a_2 > 0, "1 ", "-1"], "" + a_2) → textcoeffa2
  25. If(abs(b_2) ≟ 1, If[b_2 > 0, "1 ", "-1"], "" + b_2) → textcoeffb2
  26. If(listCoefficientsExpected ≟ listCoefficientsInputFunction, true, false) → correctfactors
  27. Textfield(textroot2i) → inputBox6
  28. FractionText((-(b_1 / a_1))) → textsolvedx1
  29. FractionText((-(b_2 / a_2))) → textsolvedx2
  30. Textfield(textroot1i) → inputBox5
  31. If(((-b_2)) / a_2 ≟ root2i, true, false) → root2iEqualx2
  32. If(((-b_1)) / a_1 ≟ root2i, true, false) → root2iEqualx1
  33. If(((-b_1)) / a_1 ≟ root1i, true, false) → root1iEqualx1
  34. If(((-b_2)) / a_2 ≟ root1i, true, false) → root1iEqualx2
  35. If(root1iEqualx1 ∧ root2iEqualx2 ∨ root2iEqualx1 ∧ root1iEqualx2, true, false) → correctroots
  36. If(abs(a_{i1}) + abs(a_{i2}) + abs(b_{i1}) + abs(b_{i2}) > 0, true, false) → attemptedsomestate
  37. Intersect(f_c, xAxis) → C, D

AI 绘图提示语参考

绘制一个二次方程的因式分解过程，包括两个一次函数f1和f2，它们相乘得到二次函数f_c，然后分解f_c得到因式形式f。在图形中显示f1和f2的根，并与f_c的根进行比较，确保根正确。
以上提示语为根据原始绘图命令整理而来，仅供参考。