{
  "id": "LO5",
  "folder": "05_LO5_Complex_Quadratic_Roots",
  "zip": "LO5_Complex_Quadratic_Roots_SLS_xAPI.zip",
  "title": "Complex Quadratic Root Explorer",
  "objective": "Complex roots of quadratic equations",
  "kind": "quadratic_roots",
  "accent": "#205493",
  "stages": [
    {
      "id": "LO5-S1",
      "level": "Pure imaginary roots",
      "prompt": "Solve z² + 3 = 0.",
      "instruction": "Choose both roots.",
      "response": {
        "type": "choice",
        "options": [
          "z=±√3",
          "z=±i√3",
          "z=3i",
          "z=−3"
        ],
        "answer": "z=±i√3"
      },
      "model": {
        "type": "roots",
        "rootsHidden": true,
        "equation": "z² = −3",
        "radius": 1.732,
        "angles": [
          90,
          270
        ]
      },
      "hint": [
        "z² = −3.",
        "Use i² = −1."
      ],
      "worked": [
        "Write −3 as 3i².",
        "Taking both square roots gives ±i√3.",
        "The roots are symmetric on the imaginary axis."
      ],
      "misconception": "Dropping i when taking the square root of a negative real number."
    },
    {
      "id": "LO5-S2",
      "level": "Quadratic formula",
      "prompt": "Solve z² − z + 1 = 0.",
      "instruction": "Choose the exact root pair.",
      "response": {
        "type": "choice",
        "options": [
          "(1±i√3)/2",
          "(−1±i√3)/2",
          "1±i",
          "±i"
        ],
        "answer": "(1±i√3)/2"
      },
      "model": {
        "type": "roots",
        "rootsHidden": true,
        "equation": "Δ = 1 − 4 = −3",
        "radius": 1,
        "angles": [
          60,
          300
        ]
      },
      "hint": [
        "The discriminant is −3.",
        "Use z=(1±√−3)/2."
      ],
      "worked": [
        "Δ=b²−4ac=−3.",
        "√Δ=i√3.",
        "Thus z=(1±i√3)/2."
      ],
      "misconception": "Losing the denominator 2a or replacing √−3 by −√3."
    },
    {
      "id": "LO5-S3",
      "level": "Complex coefficient",
      "prompt": "Solve z² + iz − 7 = 0.",
      "instruction": "Choose the exact pair.",
      "response": {
        "type": "choice",
        "options": [
          "(−i±3√3)/2",
          "(i±3√3)/2",
          "(−1±i√27)/2",
          "±√7−i"
        ],
        "answer": "(−i±3√3)/2"
      },
      "model": {
        "type": "components",
        "expression": "z=[−i ± √(i²+28)]/2",
        "real": "±3√3/2",
        "imag": "−1/2"
      },
      "hint": [
        "i² = −1, so the discriminant is 27.",
        "The −b term is −i."
      ],
      "worked": [
        "Δ=i²−4(1)(−7)=−1+28=27.",
        "√Δ=3√3.",
        "Therefore z=(−i±3√3)/2."
      ],
      "misconception": "Treating i² as +1 or putting the radical on the imaginary component."
    },
    {
      "id": "LO5-S4",
      "level": "Leading i coefficient",
      "prompt": "Solve iz² + 3z − 2i = 0.",
      "instruction": "Choose the pair after simplifying carefully.",
      "response": {
        "type": "choice",
        "options": [
          "z=i, 2i",
          "z=−i, −2i",
          "z=1,2",
          "z=±i√2"
        ],
        "answer": "z=i, 2i"
      },
      "model": {
        "type": "components",
        "expression": "multiply by −i",
        "real": "z² − 2",
        "imag": "−3iz"
      },
      "hint": [
        "Multiply the equation by −i.",
        "You obtain z²−3iz−2=0."
      ],
      "worked": [
        "Multiplying by −i gives z²−3iz−2=0.",
        "It factors as (z−i)(z−2i).",
        "Hence z=i or z=2i."
      ],
      "misconception": "Multiplying −2i by −i incorrectly or forgetting that i·i=−1."
    },
    {
      "id": "LO5-S5",
      "level": "Square a complex number",
      "prompt": "The number z satisfies z² = 3 − 4i. Find both values.",
      "instruction": "Choose the exact pair without a calculator.",
      "response": {
        "type": "choice",
        "options": [
          "z=±(2−i)",
          "z=±(2+i)",
          "z=3±4i",
          "z=±(1−2i)"
        ],
        "answer": "z=±(2−i)"
      },
      "model": {
        "type": "components",
        "expression": "(a+bi)²=3−4i",
        "real": "a²−b²=3",
        "imag": "2ab=−4"
      },
      "hint": [
        "Look for integers a,b with 2ab=−4.",
        "Check the real equation a²−b²=3."
      ],
      "worked": [
        "Choose a=2 and b=−1.",
        "(2−i)²=4−4i+i²=3−4i.",
        "Both square roots are ±(2−i)."
      ],
      "misconception": "Taking square roots component by component or returning only one root."
    },
    {
      "id": "LO5-S6",
      "level": "Reverse engineer",
      "prompt": "Which quadratic has roots 2+3i and 2−3i?",
      "instruction": "Build it from the root pair.",
      "response": {
        "type": "choice",
        "options": [
          "z²−4z+13=0",
          "z²−4z−5=0",
          "z²+4z+13=0",
          "z²−2z+9=0"
        ],
        "answer": "z²−4z+13=0"
      },
      "model": {
        "type": "roots",
        "roots": [
          {
            "re": 2,
            "im": 3
          },
          {
            "re": 2,
            "im": -3
          }
        ],
        "labels": [
          "α",
          "α*"
        ]
      },
      "hint": [
        "Use z²−(sum)z+(product).",
        "The product is 2²+3²."
      ],
      "worked": [
        "The sum is (2+3i)+(2−3i)=4.",
        "The product is (2+3i)(2−3i)=13.",
        "So the quadratic is z²−4z+13=0."
      ],
      "misconception": "Using the difference of squares 4−9 instead of the conjugate product 4+9."
    }
  ]
}