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Solve Quadratic Equations by Factorization

Solve Quadratic Equations by Factorization
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2013-05-05 16:18
Function
This example demonstrates how to solve quadratic equations using factorization. Key math concepts involved include quadratic equations, factorization, and root finding. It utilizes GeoGebra commands such as If, GCD, Simplify, Factor, Textfield, Coefficients, FractionText, Intersect, and objects like 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. Suitable for middle school and above students for classroom demonstration and self-practice.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw the process of factoring a quadratic equation, including two linear functions f1 and f2, which multiply to form the quadratic function f_c. Then factor f_c to get its factored form f. Display the roots of f1 and f2 in the graph and compare them with the roots of f_c to ensure they are correct.
The prompt above was compiled from the original drawing commands and is for reference only.