// Numbas version: finer_feedback_settings
{"name": "Quadratic Equations: The Discriminant 01", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quadratic Equations: The Discriminant 01", "tags": [], "metadata": {"description": "<p>Calculation of quadratic discriminants.</p>\n<p>State nature of roots.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h4>Using the Quadratic Formula, the roots of a quadratic equation&nbsp; \\(ax^2+ bx +c =0 \\)&nbsp; are given by:</h4>\n<p style=\"text-align: center;\">\\( \\Large x=\\frac{-b \\pm \\sqrt{ \\color{red}{b^2 - 4ac}}}{2a} \\)</p>\n<p>The discriminant (\\( \\Delta \\))&nbsp; is equal to the expression inside the square root (shown in red). It can be used to find the number and nature of the equation's roots.</p>\n<p style=\"text-align: center;\">\\( \\large \\Delta = b^2 - 4ac \\)</p>\n<p>&nbsp;If&nbsp; \\( \\Delta &gt; 0 \\) (positive) then the quadratic has two distinct, real roots,</p>\n<p>&nbsp;If&nbsp; \\( \\Delta = 0 \\) then the quadratic has one real, repeated root,</p>\n<p>&nbsp;If&nbsp; \\( \\Delta &lt; 0 \\) (negative) then the roots are both non-real (complex).</p>", "advice": "<p>We are asked to rxamine a variety of quadratic equations, calculate their discriminants and then state the number and nature of their roots,</p>\n<p></p>\n<p>a)</p>\n<p>\\( \\simplify{ x^2 + {b1}x + {c1} } =0 \\)</p>\n<p>First identify the co-efficients:&nbsp;&nbsp;&nbsp; \\( a = \\var{a1} \\),&nbsp; \\( b = \\var{b1} \\)&nbsp; and&nbsp; \\( c= \\var{c1} \\).</p>\n<p>We use the formula to calculate the discriminant:</p>\n<p>\\( \\Delta = b^2 - 4ac \\)</p>\n<p>\\( \\Delta = \\var{b1}^2 - 4 \\times \\var{a1} \\times \\var{c1} \\)</p>\n<p>\\( \\Delta = \\var{disc1} \\)</p>\n<p>Since the discriminant <strong>equals zero</strong>, we can state that this equation has a<strong><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><span class=\"student-answer\" data-bind=\"css: {answered: scoreFeedback.answered, 'has-warnings': hasWarnings}, attr: {&quot;feedback-state&quot;: scoreFeedback.state}\" feedback-state=\"none\"><span localise-data-jme-context-description=\"part.mcq.choices\"> single real, repeated root.</span></span></span></strong></p>\n<p><strong><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><span class=\"student-answer\" data-bind=\"css: {answered: scoreFeedback.answered, 'has-warnings': hasWarnings}, attr: {&quot;feedback-state&quot;: scoreFeedback.state}\" feedback-state=\"none\"><span localise-data-jme-context-description=\"part.mcq.choices\">&nbsp;</span></span></span></strong></p>\n<p><strong><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><span class=\"student-answer\" data-bind=\"css: {answered: scoreFeedback.answered, 'has-warnings': hasWarnings}, attr: {&quot;feedback-state&quot;: scoreFeedback.state}\" feedback-state=\"none\"><span localise-data-jme-context-description=\"part.mcq.choices\">&nbsp;</span></span></span></strong></p>\n<p>b)</p>\n<p>\\( \\simplify{ x^2 + {b2}x + {c2} } =0 \\)</p>\n<p>First identify the co-efficients:&nbsp;&nbsp;&nbsp; \\( a = \\var{a1} \\),&nbsp; \\( b = \\var{b2} \\)&nbsp; and&nbsp; \\( c= \\var{c2} \\).</p>\n<p>We use the formula to calculate the discriminant:</p>\n<p>\\( \\Delta = b^2 - 4ac \\)</p>\n<p>\\( \\Delta = \\var{b2}^2 - 4 \\times \\var{a1} \\times \\var{c2} \\)</p>\n<p>\\( \\Delta = \\var{disc2} \\)</p>\n<p>Since the discriminant is <strong>positive</strong>, we can state that this equation has <strong>t<span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><span class=\"student-answer\" data-bind=\"css: {answered: scoreFeedback.answered, 'has-warnings': hasWarnings}, attr: {&quot;feedback-state&quot;: scoreFeedback.state}\" feedback-state=\"none\"><span localise-data-jme-context-description=\"part.mcq.choices\"><strong>wo</strong> distinct, real roots.</span></span></span></strong></p>\n<p><strong><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><span class=\"student-answer\" data-bind=\"css: {answered: scoreFeedback.answered, 'has-warnings': hasWarnings}, attr: {&quot;feedback-state&quot;: scoreFeedback.state}\" feedback-state=\"none\"><span localise-data-jme-context-description=\"part.mcq.choices\"></span></span></span></strong></p>\n<p><strong><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><span class=\"student-answer\" data-bind=\"css: {answered: scoreFeedback.answered, 'has-warnings': hasWarnings}, attr: {&quot;feedback-state&quot;: scoreFeedback.state}\" feedback-state=\"none\"><span localise-data-jme-context-description=\"part.mcq.choices\">&nbsp;</span></span></span></strong></p>\n<p><strong><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><span class=\"student-answer\" data-bind=\"css: {answered: scoreFeedback.answered, 'has-warnings': hasWarnings}, attr: {&quot;feedback-state&quot;: scoreFeedback.state}\" feedback-state=\"none\"><span localise-data-jme-context-description=\"part.mcq.choices\">&nbsp;</span></span></span></strong></p>\n<p>c)</p>\n<p>\\( \\simplify{ x^2 + {b3}x + {c3} } =0 \\)</p>\n<p>First identify the co-efficients:&nbsp;&nbsp;&nbsp; \\( a = \\var{a1} \\),&nbsp; \\( b = \\var{b3} \\)&nbsp; and&nbsp; \\( c= \\var{c3} \\).</p>\n<p>We use the formula to calculate the discriminant:</p>\n<p>\\( \\Delta = b^2 - 4ac \\)</p>\n<p>\\( \\Delta = \\var{b3}^2 - 4 \\times \\var{a1} \\times \\var{c3} \\)</p>\n<p>\\( \\Delta = \\var{disc3} \\)</p>\n<p>Since the discriminant is <strong>negative</strong>, we can state that this equation has <strong>t<span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><span class=\"student-answer\" data-bind=\"css: {answered: scoreFeedback.answered, 'has-warnings': hasWarnings}, attr: {&quot;feedback-state&quot;: scoreFeedback.state}\" feedback-state=\"none\"><span localise-data-jme-context-description=\"part.mcq.choices\">wo distinct, non-real (complex) roots.</span></span></span></strong></p>\n<p><strong><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><span class=\"student-answer\" data-bind=\"css: {answered: scoreFeedback.answered, 'has-warnings': hasWarnings}, attr: {&quot;feedback-state&quot;: scoreFeedback.state}\" feedback-state=\"none\"><span localise-data-jme-context-description=\"part.mcq.choices\">&nbsp;</span></span></span></strong></p>\n<p><strong><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><span class=\"student-answer\" data-bind=\"css: {answered: scoreFeedback.answered, 'has-warnings': hasWarnings}, attr: {&quot;feedback-state&quot;: scoreFeedback.state}\" feedback-state=\"none\"><span localise-data-jme-context-description=\"part.mcq.choices\">&nbsp;</span></span></span></strong></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"f1": {"name": "f1", "group": "Repeated roots", "definition": "random(-9..9 except -1 except 0 except 1)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Repeated roots", "definition": "{f1}+{f1}", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Repeated roots", "definition": "{f1}*{f1}", "description": "", "templateType": "anything", "can_override": false}, "disc1": {"name": "disc1", "group": 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"{b2}^2-(4*{a1}*{c2})", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Complex roots", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Complex roots", "definition": "random(-9..9 except -1 except 0 except 1)", "description": "", "templateType": "anything", "can_override": false}, "cr1": {"name": "cr1", "group": "Complex roots", "definition": "{p}+{q}i", "description": "", "templateType": "anything", "can_override": false}, "cr2": {"name": "cr2", "group": "Complex roots", "definition": "{p}-{q}i", "description": "", "templateType": "anything", "can_override": false}, "c3": {"name": "c3", "group": "Complex roots", "definition": "{cr1}*{cr2}", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Complex roots", "definition": "{cr1}+{cr2}", "description": "", "templateType": "anything", "can_override": false}, "disc3": {"name": "disc3", "group": "Complex roots", "definition": "{b3}^2-(4*{a1}*{c3})", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Repeated roots", "variables": ["f1", "a1", "b1", "c1", "disc1"]}, {"name": "Two real roots", "variables": ["g1", "g2", "b2", "c2", "disc2"]}, {"name": "Complex roots", "variables": ["p", "q", "cr1", "cr2", "b3", "c3", "disc3"]}, {"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null}, {"type": "information", "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": "<p>For the following quadratic equations, calculate their discriminant. Then determine the number and nature of the roots for each:</p>"}, {"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{ x^2 + {b1}x + {c1} } =0 \\)</p>\n<p>The discriminant of this equation:&nbsp;&nbsp; \\( \\Delta = \\)&nbsp; [[0]]</p>\n<p>This equation has roots that are:</p>\n<p>[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{disc1}", "maxValue": "{disc1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Two distinct, real roots", "A single real, repeated root", "Two distinct, non-real (complex) roots"], "matrix": ["0", "1", 0], "distractors": ["", "", ""]}], "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{ x^2 + {b2}x + {c2} } =0 \\)</p>\n<p>The discriminant of this equation:&nbsp;&nbsp; \\( \\Delta = \\)&nbsp; [[0]]</p>\n<p>This equation has roots that are:</p>\n<p>[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{disc2}", "maxValue": "{disc2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Two distinct, real roots", "A single real, repeated root", "Two distinct, non-real (complex) roots"], "matrix": ["1", "0", 0], "distractors": ["", "", ""]}], "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{ x^2 + {b3}x + {c3} } =0 \\)</p>\n<p>The discriminant of this equation:&nbsp;&nbsp; \\( \\Delta = \\)&nbsp; [[0]]</p>\n<p>This equation has roots that are:</p>\n<p>[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{disc3}", "maxValue": "{disc3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Two distinct, real roots", "A single real, repeated root", "Two distinct, non-real (complex) roots"], "matrix": ["0", "0", "1"], "distractors": ["", "", ""]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}