Transcript: Solving_quadratic_equation_by_factorization_20250701_050954.html

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  {
    "slide": 1,
    "fragments": [
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        "text_description": "Quadratic Expression",
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      {
        "fragment_index": 1,
        "text_description": "Quadratic Expression",
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      },
      {
        "fragment_index": 2,
        "text_description": "An algebraic expression whose highest power of the variable is\n2\n.\nStandard form: \\(ax^{2}+bx+c\\) where \\(a\\neq 0\\).",
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      }
    ]
  },
  {
    "slide": 2,
    "fragments": [
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        "fragment_index": -1,
        "text_description": "Quadratic Equation",
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      {
        "fragment_index": 1,
        "text_description": "Quadratic Equation\nA quadratic equation is formed by setting a quadratic expression to zero: \\(ax^2 + bx + c = 0\\). The values of \\(x\\) that satisfy it are called roots; they are where the parabola crosses the x-axis.",
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      {
        "fragment_index": 2,
        "text_description": "Key Characteristics:\nStandard form \\(ax^2 + bx + c = 0\\) with \\(a \\neq 0\\).\nDegree 2; graph is a parabola.\nRoots make the expression value \\(0\\).",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Example:\n\\(x^2 - 5x + 6 = 0\\) → roots \\(x = 2, 3\\).",
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      }
    ]
  },
  {
    "slide": 3,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Why Factorise?\nFactorisation + Zero-Product Property lets us read roots instantly.",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "1\nRecall Zero-Product Property\nIf \\(AB = 0\\), at least one factor must be zero.",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "2\nTurn Quadratic into Factors\nRewrite \\(ax^{2}+bx+c\\) as \\((A)(B)=0\\) by factorising.",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "3\nSet Each Factor to Zero\nSolve \\(A = 0\\) and \\(B = 0\\) to get both roots quickly.",
        "image_description": ""
      },
      {
        "fragment_index": 4,
        "text_description": "Pro Tip:\nLook for factor pairs of \\(a \\times c\\) to spot brackets fast.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 4,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Split the Middle\nMiddle-term splitting helps factorise quadratics when \\(a = 1\\).",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "1\nFind the Pair\nChoose two numbers whose product equals \\(c\\) and sum equals \\(b\\).",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "2\nSplit the Middle Term\nRewrite \\(bx\\) as \\(mx + nx\\) using those numbers.",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "3\nGroup & Factor\nGroup terms in pairs and factor their GCFs.",
        "image_description": ""
      },
      {
        "fragment_index": 4,
        "text_description": "4\nForm Brackets & Solve\nWrite the common bracket, set each factor to zero, solve for \\(x\\).",
        "image_description": ""
      },
      {
        "fragment_index": 5,
        "text_description": "Pro Tip:\nExample—\\(x^{2}+7x+12\\): numbers are 3 and 4 because \\(3\\times4=12\\) and \\(3+4=7\\).",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 5,
    "fragments": [
      {
        "fragment_index": 1,
        "text_description": "Solved Examples\nExample 1 – Easy coefficients",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "Problem Statement\nSolve \\(x^{2}+5x+6=0\\) by factorisation.",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Solution Process\n1\nPick the numbers\nProduct \\(=6\\), sum \\(=5\\). Choose \\(2\\) and \\(3\\).\n\\(2 \\times 3 = 6,\\; 2 + 3 = 5\\)\n2\nSplit the middle term\nRewrite the quadratic using \\(2\\) and \\(3\\).\n\\(x^{2}+2x+3x+6=0\\)\nKey Insight\nChoosing the right pair of numbers makes factorisation quick and error-free.\n3\nFactorise and solve\nGroup terms, extract the common bracket, then apply the zero-product rule.\n\\(x(x+2)+3(x+2)=0\\\\(x+2)(x+3)=0\\\\x=-2 \\text{ or } -3\\)",
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      },
      {
        "fragment_index": 4,
        "text_description": "Final Answer\nRoots: \\(x=-2\\) or \\(x=-3\\)\nBoth values satisfy \\(x^{2}+5x+6=0\\).",
        "image_description": ""
      },
      {
        "fragment_index": 5,
        "text_description": "Additional Notes\nCommon Mistakes\nPicking \\(1\\) and \\(6\\); sum is not \\(5\\).\nNot setting each factor equal to zero.\nRelated Topics\nZero product property\nQuadratic formula",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 6,
    "fragments": [
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        "fragment_index": -1,
        "text_description": "Multiple Choice Question\nQuestion\nWhich factorised form solves \\(x^{2}+7x+12=0\\)?\nSubmit Answer\nCorrect!\nGreat! \\(4+3=7\\) and \\(4\\\\times3=12\\).\nIncorrect\nRecheck the sum and product of your chosen pair.\nconst correctOption = 0;\n    const answerCards = document.querySelectorAll('.answer-card');\n    const submitBtn = document.getElementById('submitBtn');\n    const feedbackCorrect = document.getElementById('feedbackCorrect');\n    const feedbackIncorrect = document.getElementById('feedbackIncorrect');\n\n    let selectedOption = null;\n\n    answerCards.forEach((card, index) => {\n      card.addEventListener('click', () => {\n        answerCards.forEach(c => c.classList.remove('border-blue-500', 'bg-blue-50'));\n        card.classList.add('border-blue-500', 'bg-blue-50');\n        selectedOption = index;\n      });\n    });\n\n    submitBtn.addEventListener('click', () => {\n      if (selectedOption === null) return;\n\n      if (selectedOption === correctOption) {\n        feedbackCorrect.classList.remove('hidden');\n        feedbackIncorrect.classList.add('hidden');\n      } else {\n        feedbackIncorrect.classList.remove('hidden');\n        feedbackCorrect.classList.add('hidden');\n      }\n    });",
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      {
        "fragment_index": 1,
        "text_description": "A\n\\((x+4)(x+3)=0\\)",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "B\n\\((x+6)(x+1)=0\\)",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "C\n\\((x+5)(x+2)=0\\)",
        "image_description": ""
      },
      {
        "fragment_index": 4,
        "text_description": "D\n\\((x+12)(x-1)=0\\)",
        "image_description": ""
      },
      {
        "fragment_index": 5,
        "text_description": "Hint:\nFind two numbers that add to 7 and multiply to 12.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 7,
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        "text_description": "Order the Steps\nDraggable Items\nDrop Zones\nResults\n// Drag and drop functionality\n    const draggableItems = document.querySelectorAll('.draggable-item');\n    const dropZones = document.querySelectorAll('.drop-zone');\n    const checkAnswersBtn = document.getElementById('checkAnswersBtn');\n    const feedbackArea = document.getElementById('feedbackArea');\n    const feedbackContent = document.getElementById('feedbackContent');\n\n    draggableItems.forEach(item => {\n      item.addEventListener('dragstart', handleDragStart);\n      item.addEventListener('dragend', handleDragEnd);\n    });\n\n    dropZones.forEach(zone => {\n      zone.addEventListener('dragover', handleDragOver);\n      zone.addEventListener('drop', handleDrop);\n      zone.addEventListener('dragenter', handleDragEnter);\n      zone.addEventListener('dragleave', handleDragLeave);\n    });\n\n    function handleDragStart(e) {\n      e.target.classList.add('opacity-50');\n      e.dataTransfer.setData('text/plain', e.target.dataset.id);\n    }\n\n    function handleDragEnd(e) {\n      e.target.classList.remove('opacity-50');\n    }\n\n    function handleDragOver(e) {\n      e.preventDefault();\n    }\n\n    function handleDragEnter(e) {\n      e.preventDefault();\n      e.target.closest('.drop-zone').classList.add('border-green-500', 'bg-green-50');\n    }\n\n    function handleDragLeave(e) {\n      e.target.closest('.drop-zone').classList.remove('border-green-500', 'bg-green-50');\n    }\n\n    function handleDrop(e) {\n      e.preventDefault();\n      const dropZone = e.target.closest('.drop-zone');\n      dropZone.classList.remove('border-green-500', 'bg-green-50');\n\n      const itemId = e.dataTransfer.getData('text/plain');\n      const draggedItem = document.querySelector(`[data-id=\"${itemId}\"]`);\n\n      if (draggedItem && dropZone) {\n        dropZone.appendChild(draggedItem);\n        dropZone.querySelector('.text-center').style.display = 'none';\n      }\n    }\n\n    // Check answers functionality\n    checkAnswersBtn.addEventListener('click', () => {\n      const correctOrder = ['item1', 'item2', 'item3', 'item4'];\n      const userOrder = [];\n\n      dropZones.forEach(zone => {\n        const child = zone.querySelector('.draggable-item');\n        if (child) userOrder.push(child.dataset.id);\n      });\n\n      feedbackArea.classList.remove('hidden');\n\n      if (JSON.stringify(correctOrder) === JSON.stringify(userOrder)) {\n        feedbackContent.innerHTML = '<p class=\"text-green-600 font-semibold\">Great job! The steps are in the correct order.</p>';\n      } else {\n        feedbackContent.innerHTML = '<p class=\"text-red-600 font-semibold\">Some steps are misplaced. Try again!</p>';\n      }\n    });",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "Arrange the four steps needed to factorise \\(x^{2}+8x+15\\) by middle-term splitting.",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "Find two numbers: product = 15, sum = 8",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Rewrite \\(8x\\) using those numbers",
        "image_description": ""
      },
      {
        "fragment_index": 4,
        "text_description": "Group the four terms in pairs",
        "image_description": ""
      },
      {
        "fragment_index": 5,
        "text_description": "Take the common bracket to get factors",
        "image_description": ""
      },
      {
        "fragment_index": 6,
        "text_description": "Step 1",
        "image_description": ""
      },
      {
        "fragment_index": 7,
        "text_description": "Step 2",
        "image_description": ""
      },
      {
        "fragment_index": 8,
        "text_description": "Step 3",
        "image_description": ""
      },
      {
        "fragment_index": 9,
        "text_description": "Step 4",
        "image_description": ""
      },
      {
        "fragment_index": 10,
        "text_description": "Tip:\nRemember: 5 × 3 = 15 and 5 + 3 = 8.",
        "image_description": ""
      },
      {
        "fragment_index": 11,
        "text_description": "Check Answers",
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  },
  {
    "slide": 8,
    "fragments": [
      {
        "fragment_index": 1,
        "text_description": "Example 2\nCoefficient a ≠ 1  |  Factorisation Method",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "Problem Statement\nSolve \\(2x^{2}+7x+3=0\\) by factorisation.",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Solution Process\n1\nMultiply \\(a\\) and \\(c\\)\n\\(a \\times c = 2 \\times 3 = 6\\).\n\\(ac = 6\\)\n2\nSplit the middle term\nNumbers with product 6 and sum 7 are 6 and 1.\n                                Rewrite as \\(2x^{2}+6x+x+3=0\\).\n\\(2x^{2}+6x+x+3=0\\)\nKey Insight\nFor \\(a>1\\), always start with \\(ac\\). The splitting method then works just like when \\(a=1\\).\n3\nGroup and factor\n\\(2x(x+3)+1(x+3)=0 \\Rightarrow (x+3)(2x+1)=0\\).\n\\((x+3)(2x+1)=0\\)",
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      {
        "fragment_index": 4,
        "text_description": "Final Answer\nRoots: \\(x=-3\\) or \\(x=-\\\\dfrac{1}{2}\\)\nFactorisation complete; learning outcome achieved.",
        "image_description": ""
      },
      {
        "fragment_index": 5,
        "text_description": "Additional Notes\nCommon Mistakes\nSkipping \\(ac\\) multiplication.\nLosing a common factor during grouping.\nRelated Topics\nQuadratic formula.\nCompleting the square.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 9,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Key Takeaways\nThank You!\nWe hope you found this lesson informative and engaging.",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "Write the quadratic as \\(ax^{2}+bx+c=0\\).",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "Factorise into two brackets \\( (A)(B)=0 \\) and set each to zero.",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Split the middle term: numbers multiply to \\(c\\) and add to \\(b\\).",
        "image_description": ""
      },
      {
        "fragment_index": 4,
        "text_description": "If \\(a\\neq1\\), use the product \\(a\\times c\\) instead.",
        "image_description": ""
      },
      {
        "fragment_index": 5,
        "text_description": "Expand to verify the factors match the original equation.",
        "image_description": ""
      },
      {
        "fragment_index": 6,
        "text_description": "Next Steps\nPractise on varied quadratics; test both \\(a=1\\) and \\(a\\neq1\\), then check roots by substitution.",
        "image_description": ""
      }
    ]
  }
]