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{"name": "Solution of Quadratic Equation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Solution of Quadratic Equation", "tags": [], "metadata": {"description": "This is a new and easy way for solving quadratic equation. It is based on the property that the roots are equidistant from the average.&nbsp;", "licence": "None specified"}, "statement": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<h2><span style=\"text-decoration: underline;\"><strong>A new way for solving Quadratic Equation:</strong></span></h2>\n<p>The quadratic equation $ax^2+bx+c=0$ can be solved in a different and easy way as follows:</p>\n<p>Let $\\alpha$ and $\\beta$ be the two roots.&nbsp; Then we know $\\alpha\\times\\beta=\\dfrac{c}{a}$ and $\\alpha+\\beta=-\\dfrac{b}{a}$</p>\n<p>We shall use the property that the roots are equidistant from their average.</p>\n<p>So we can take them as $\\alpha=\\dfrac{1}{2}\\rm{sum}-u$ and $\\beta=\\dfrac{1}{2}\\rm{sum}+u$ Or $\\boxed{\\alpha=-\\dfrac{1}{2}\\dfrac{b}{a}-u}$ and&nbsp;$\\boxed{\\beta=-\\dfrac{1}{2}\\dfrac{b}{a}+u}$ .</p>\n<p>Consequently we have $\\alpha\\times\\beta=\\left(-\\dfrac{1}{2}\\dfrac{b}{a}-u\\right)\\times\\left(-\\dfrac{1}{2}\\dfrac{b}{a}+u\\right)$</p>\n<p>$\\implies \\dfrac{c}{a}=\\dfrac{1}{4}\\dfrac{b^2}{a^2}-u^2\\implies u^2=\\dfrac{1}{4}\\dfrac{b^2}{a^2}-\\dfrac{c}{a}$.</p>\n<p>Simplifying $u^2=\\dfrac{b^2-4ac}{4a^2}\\implies \\boxed{u=\\dfrac{\\sqrt{b^2-4ac}}{2a}}$ .</p>\n<p>Substituting above we get</p>\n<p>$\\alpha=\\dfrac{-b-\\sqrt{b^2-4ac}}{2a}$ and $\\beta=\\dfrac{-b+\\sqrt{b^2-4ac}}{2a}$.</p>\n<p>Note : $\\boxed{\\alpha=\\rm{average}-u}$ and $\\boxed{\\beta=\\rm{average}+u}$</p>\n<p>______________________________________________________________________________________________________________________________</p>\n<p>Let us consider the following example:</p>\n<p>$x^2-5x+6=0$ .</p>\n<p>We generally start from the product $6$ and find factors by inspection so that the sum will be $5$. In stead, in this method we start from the sum and find $u$.</p>\n<p>Now sum of roots$=5\\implies \\rm{average}=\\dfrac{5}{2}$.</p>\n<p>So roots are $\\dfrac{5}{2}-u$ and&nbsp;$\\dfrac{5}{2}+u$ .</p>\n<p>Therefore product of roots $=\\dfrac{25}{4}-u^2\\implies 6=\\dfrac{25}{4}-u^2\\implies u^2=\\dfrac{25}{4}-6\\implies u^2=\\dfrac{1}{4}\\implies u=\\pm\\dfrac{1}{2}$.</p>\n<p>So roots are $\\dfrac{5}{2}-\\dfrac{1}{2}$ and $\\dfrac{5}{2}+\\dfrac{1}{2}$ .&nbsp;Or roots are $2$ and $3$.&nbsp;</p>\n<p>______________________________________________________________________________________________________________________________&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</p>", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..5)*random(1,-1)", "description": "<div id=\"MathJax_Message\" style=\"display: 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none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Distance of the root from the average</p>", "templateType": "anything", "can_override": false}, "avg": {"name": "avg", "group": "Ungrouped variables", "definition": "rational_approximation(sum[0]/sum[1]*0.5)", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Average of the roots</p>", "templateType": "anything", "can_override": false}, "alpha": {"name": "alpha", "group": "Ungrouped variables", "definition": "rational_approximation((avg[0]-(u[0]/u[1])*avg[1])/avg[1])", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Average-u</p>", "templateType": "anything", "can_override": false}, "beta": {"name": "beta", "group": "Ungrouped variables", "definition": "rational_approximation((avg[0]+(u[0]/u[1])*avg[1])/avg[1])", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div 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100}, "ungrouped_variables": ["a", "n", "b", "c", "dis", "sum", "product", "avg", "u", "alpha", "beta"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<h2>PRACTICE EXERCISE:</h2>\n<h3>Solve the equation $\\simplify{{a}x^2+{b}x+{c}=0}$</h3>\n<p>Fill in the gaps below.</p>\n<p>a=[[0]]</p>\n<p>b=[[1]]</p>\n<p>c=[[2]]</p>\n<p>Sum of roots=[[3]]</p>\n<p>Avg(Average of roots)=[[4]]</p>\n<p>$b^2$=[[5]]</p>\n<p>$4ac$=[[6]]</p>\n<p>Discriminant 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