// Numbas version: exam_results_page_options
{"name": "Quadratic Equations: Factor and Solve (Simple Case) 01", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quadratic Equations: Factor and Solve (Simple Case) 01", "tags": [], "metadata": {"description": "<p>Questions asking for the factoring of simple case (a=1)</p>\n<p>Then using factors to find roots.</p>\n<p></p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h4>A quadratic equation in \"Standard\" form looks like this:</h4>\n<h3 style=\"text-align: center;\">\\( \\color{red}{a} x^2 + \\color{red}{b} x + \\color{red}{c} = 0 \\)</h3>\n<p>Where \\( a, b \\) and \\( c \\)&nbsp; are numerical co-efficients. \"\\( a \\)\" cannot be \\( 0 \\)&nbsp; &nbsp;but&nbsp; \"\\( b \\)\" and \"\\( c \\)\" can be any value (including \\( 0 \\)). The equation MUST be in this form before applying ANY method for solving.</p>\n<p>Some quadratic equations will factorise giving a simple way to \"solve\" them and find their roots.&nbsp;In the \"<strong>simple case</strong>\"&nbsp; (when \\( a=1 \\)) this can be acheived by finding factors whose product (\\( \\times \\)) is \\( c \\) and sum (\\( + \\)) is \\( b \\).</p>", "advice": "<p>We are asked to factor different quadratic equations.</p>\n<p>The first step is to check that the equations are in \"standard form\", in these questions, they are. Also, these equations are all \"simple case\" quadratics (the co-efficient of \\( x^2 \\) is one), i.e. \\( a=1 \\).</p>\n<p>We know that, to get the \\( x^2 \\) term we need a single&nbsp; \\( x \\) in each bracket:</p>\n<p>\\(&nbsp; ( x + ??)( x + ?? )&nbsp; \\)</p>\n<p>Then, as stated earlier, you just need to find a pair of numbers that:</p>\n<ul>\n<li>Multiply to give&nbsp; \\( c \\)</li>\n<li>Add to give&nbsp; \\(b \\).</li>\n</ul>\n<p>Once the equation is factored, we have&nbsp; \\(&nbsp; ( x + ??)( x + ?? ) =0 \\)</p>\n<p>The only way that this is possible is if one (or both) of the brackets equals zero.</p>\n<hr/>\n<p></p>\n<p>a)&nbsp; &nbsp; &nbsp;\\( \\simplify{&nbsp; x^2 + {b1}x + {c1} =0&nbsp; } \\)</p>\n<p style=\"padding-left: 40px;\">We know that, to get the \\( x^2 \\) term we need a single&nbsp; \\( x \\) in each bracket:</p>\n<p style=\"padding-left: 40px;\">\\(&nbsp; ( x + ??)( x + ?? ) =0&nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">Then, we need to find two numbers that:</p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>Multiply to give&nbsp; \\( \\var{c1} \\)</li>\n<li>Add to give&nbsp; \\( \\var{b1} \\).</li>\n</ul>\n</li>\n</ul>\n<p style=\"padding-left: 40px;\">A little thought leads us to&nbsp; \\( \\var{f1} \\)&nbsp; and&nbsp; \\( \\var{f2} \\). these can be inserted into the brackets to give:</p>\n<p style=\"padding-left: 40px;\">\\(&nbsp; \\simplify{ ( x + {f1})( x + {f2} ) }=0&nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">Now</p>\n<p style=\"padding-left: 40px;\">If&nbsp; \\(&nbsp; \\simplify{ ( x + {f1}) }=0&nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">Then&nbsp; \\(&nbsp; \\simplify{ x&nbsp; =-{f1} } \\)</p>\n<p style=\"padding-left: 40px;\">If&nbsp; \\(&nbsp; \\simplify{ ( x + {f2}) }=0&nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">Then&nbsp; \\(&nbsp; \\simplify{ x&nbsp; =-{f2} } \\)</p>\n<p>&nbsp;&nbsp;</p>\n<p>&nbsp;&nbsp;</p>\n<p>b)&nbsp; &nbsp; &nbsp;\\( \\simplify{&nbsp; x^2 + {b2}x + {c2} =0&nbsp; } \\)</p>\n<p style=\"padding-left: 40px;\">We know that, to get the \\( x^2 \\) term we need a single&nbsp; \\( x \\) in each bracket:</p>\n<p style=\"padding-left: 40px;\">\\(&nbsp; ( x + ??)( x + ?? ) =0&nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">Then, we need to find two numbers that:</p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>Multiply to give&nbsp; \\( \\var{c2} \\)</li>\n<li>Add to give&nbsp; \\( \\var{b2} \\).</li>\n</ul>\n</li>\n</ul>\n<p style=\"padding-left: 40px;\">A little thought leads us to&nbsp; \\( \\var{g1} \\)&nbsp; and&nbsp; \\( \\var{g2} \\). these can be inserted into the brackets to give:</p>\n<p style=\"padding-left: 40px;\">\\( \\simplify{&nbsp; ( x + {g1})( x + {g2} ) =0 } \\)</p>\n<p style=\"padding-left: 40px;\">Now</p>\n<p style=\"padding-left: 40px;\">If&nbsp; \\(&nbsp; \\simplify{ ( x + {g1}) }=0&nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">Then&nbsp; \\(&nbsp; \\simplify{ x&nbsp; =-{g1} } \\)</p>\n<p style=\"padding-left: 40px;\">If&nbsp; \\(&nbsp; \\simplify{ ( x + {g2}) }=0&nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">Then&nbsp; \\(&nbsp; \\simplify{ x&nbsp; =-{g2} } \\)</p>\n<p style=\"padding-left: 40px;\">&nbsp; &nbsp;</p>\n<p style=\"padding-left: 40px;\">&nbsp;</p>\n<p>c)&nbsp; &nbsp; &nbsp;\\( \\simplify{&nbsp; x^2 + {b3}x + {c3} =0&nbsp; } \\)</p>\n<p style=\"padding-left: 40px;\">We know that, to get the \\( x^2 \\) term we need a single&nbsp; \\( x \\) in each bracket:</p>\n<p style=\"padding-left: 40px;\">\\(&nbsp; ( x + ??)( x + ?? ) =0&nbsp; \\)</p>\n<p style=\"padding-left: 40px;\">Then, we need to find two numbers that:</p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>Multiply to give&nbsp; \\( \\var{c3} \\)</li>\n<li>Add to give&nbsp; \\( \\var{b3} \\).</li>\n</ul>\n</li>\n</ul>\n<p style=\"padding-left: 40px;\">A little thought leads us to&nbsp; \\( \\var{h1} \\)&nbsp; and&nbsp; \\( \\var{h2} \\). these can be inserted into the brackets to give:</p>\n<p style=\"padding-left: 40px;\">\\( \\simplify{&nbsp; ( x + {h1})( x + {h2} ) =0 } \\)</p>\n<p style=\"padding-left: 40px;\">Now</p>\n<p style=\"padding-left: 40px;\">If&nbsp; 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x^2 + {b2}x + {c2} =0&nbsp; } \\)</p>\n<p>&nbsp;[[0]] \\( =0 \\)</p>\n<p>Now use these factors to give the roots of the equation:</p>\n<p>\\(x_1= \\)&nbsp; [[1]]</p>\n<p>\\(x_2= \\)&nbsp; [[2]]</p>\n<p><sub><em>(If you are sure that your roots are correct but they are marked wrong, switch them around).</em></sub></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(x + {g1} ) (x + {g2} )", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{g1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{g2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>\\( \\simplify{&nbsp; 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