{
  "schema_version": 1,
  "prompt": {
    "type": "doc",
    "blocks": [
      {
        "type": "paragraph",
        "content": [
          { "type": "text", "text": "Consider the parabola given by " },
          { "type": "math", "latex": "y = x^2 - 6x + 5" },
          { "type": "text", "text": ". What are the coordinates of its vertex?" }
        ]
      }
    ]
  },
  "answer": {
    "type": "multiple_choice",
    "shuffle": true,
    "min_selections": 1,
    "max_selections": 1,
    "choices": [
      {
        "id": "a",
        "correct": true,
        "content": {
          "type": "doc",
          "blocks": [
            {
              "type": "paragraph",
              "content": [{ "type": "math", "latex": "(3, -4)" }]
            }
          ]
        }
      },
      {
        "id": "b",
        "correct": false,
        "content": {
          "type": "doc",
          "blocks": [
            {
              "type": "paragraph",
              "content": [{ "type": "math", "latex": "(-3, -4)" }]
            }
          ]
        }
      },
      {
        "id": "c",
        "correct": false,
        "content": {
          "type": "doc",
          "blocks": [
            {
              "type": "paragraph",
              "content": [{ "type": "math", "latex": "(6, 5)" }]
            }
          ]
        }
      },
      {
        "id": "d",
        "correct": false,
        "content": {
          "type": "doc",
          "blocks": [
            {
              "type": "paragraph",
              "content": [{ "type": "math", "latex": "(3, 4)" }]
            }
          ]
        }
      }
    ]
  },
  "hints": [
    {
      "type": "doc",
      "blocks": [
        {
          "type": "paragraph",
          "content": [
            { "type": "text", "text": "The x-coordinate of the vertex of " },
            { "type": "math", "latex": "y = ax^2 + bx + c" },
            { "type": "text", "text": " is " },
            { "type": "math", "latex": "x = -\\frac{b}{2a}" },
            { "type": "text", "text": "." }
          ]
        }
      ]
    },
    {
      "type": "doc",
      "blocks": [
        {
          "type": "paragraph",
          "content": [
            {
              "type": "text",
              "text": "Once you know the x-coordinate, substitute it back into the equation to find the y-coordinate."
            }
          ]
        }
      ]
    }
  ],
  "solution": {
    "type": "doc",
    "blocks": [
      {
        "type": "paragraph",
        "content": [
          { "type": "text", "text": "Here " },
          { "type": "math", "latex": "a = 1" },
          { "type": "text", "text": " and " },
          { "type": "math", "latex": "b = -6" },
          { "type": "text", "text": ", so the vertex lies at" }
        ]
      },
      {
        "type": "math_block",
        "latex": "x = -\\frac{b}{2a} = -\\frac{-6}{2 \\cdot 1} = 3."
      },
      {
        "type": "paragraph",
        "content": [
          { "type": "text", "text": "Substituting back:" }
        ]
      },
      {
        "type": "math_block",
        "latex": "y = 3^2 - 6 \\cdot 3 + 5 = 9 - 18 + 5 = -4."
      },
      {
        "type": "paragraph",
        "content": [
          { "type": "text", "text": "The vertex is " },
          { "type": "math", "latex": "(3, -4)" },
          {
            "type": "text",
            "text": ". Equivalently, completing the square gives "
          },
          { "type": "math", "latex": "y = (x - 3)^2 - 4" },
          { "type": "text", "text": ", which reads off the same vertex directly." }
        ]
      }
    ]
  }
}