// Numbas version: exam_results_page_options
{"name": "Quadratic Equations: The Quadratic Formula (Instructional)", "metadata": {"description": "<p>Identify co-efficients, then use quadratic formula to find roots.</p>\n<p>All set to be distinct and real. Some can be non-integer (0.5 steps)</p>", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": "80", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", ""], "variable_overrides": [[], []], "questions": [{"name": "Quadratic Equations: The Quadratic Formula 01", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "<p>Solve quadratic equations (non-simple case, two real, discrete integer roots) using the formula.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h4>If we have a quadratic equation of the form&nbsp; \\( ax^2 + bx +c = 0 \\),&nbsp; we can use the Quadratic Formula:</h4>\n<p>&nbsp;</p>\n<p style=\"text-align: center;\">\\( \\LARGE x= \\frac{-b \\pm \\sqrt{b^2 -4ac}}{2a} \\)</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>to find the roots of the equation. (Note the plus-or-minus means there are two solutions, not just one.)</p>\n<p>It is also sometimes known as Shreedhara Acharya's formula, named after the 10<sup>th</sup> century Indian mathematician who first derived it.</p>", "advice": "<p>We are asked to find the roots of various quadratic equations using the Quadratic Formula. To enable us to do this, we are first asked to identify the co-efficients&nbsp; \\( a \\), \\( b \\) and \\( c \\) each time.</p>\n<p></p>\n<p>a)</p>\n<p>\\( \\simplify{ {a1}x^2 + {b1}x + {c1} } =0 \\)</p>\n<p>The co-efficients of this equation can simply be read from the equation, taking care to take account of positive and negative values:</p>\n<p>\\( a = \\)&nbsp; \\( \\var{a1}\\) &nbsp;&nbsp;&nbsp; \\( b = \\)&nbsp; \\( \\var{b1}\\) &nbsp; and&nbsp; \\( c = \\)&nbsp; \\( \\var{c1}\\)</p>\n<p>We now write the equation substituting these values in the correct places and, again, being careful with any negative values:</p>\n<p>\\( x= \\frac{-b \\pm \\sqrt{b^2 -4ac}}{2a} \\)&nbsp;&nbsp;&nbsp;&nbsp; becomes&nbsp;&nbsp;&nbsp;&nbsp; \\( x= \\frac{-(\\var{b1}) \\pm \\sqrt{(\\var{b1})^2 -4 \\times (\\var{a1}) \\times (\\var{c1})}}{2 \\times (\\var{a1})} \\)</p>\n<p>Simplify this as much as possible (mainly the expression inside the square root and the denominator):</p>\n<p>\\( x= \\frac{\\simplify{-{b1}} \\pm \\sqrt{\\simplify{{b1}^2 -4*{a1}*{c1}}}}{\\simplify{2{a1}}} \\)</p>\n<p>Now, carry out this calculation twice, first using the plus (\\( + \\))&nbsp; and then using the minus (\\( - \\)).</p>\n<p>This will give you the two roots:&nbsp;&nbsp;&nbsp; \\( x = \\var{x1} \\)&nbsp; and &nbsp; \\( x = \\var{x2} \\)</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p>b)</p>\n<p>\\( \\simplify{ {a2}x^2 + {b2}x + {c2} } =0 \\)</p>\n<p>The co-efficients of this equation can simply be read from the equation, taking care to take account of positive and negative values:</p>\n<p>\\( a = \\)&nbsp; \\( \\var{a2}\\) &nbsp; &nbsp; \\( b = \\)&nbsp; \\( \\var{b2}\\) &nbsp; and&nbsp; \\( c = \\)&nbsp; \\( \\var{c2}\\)</p>\n<p>We now write the equation substituting these values in the correct places and, again, being careful with any negative values:</p>\n<p>\\( x= \\frac{-b \\pm \\sqrt{b^2 -4ac}}{2a} \\)&nbsp;&nbsp;&nbsp;&nbsp; becomes&nbsp;&nbsp;&nbsp;&nbsp; \\( x= \\frac{-(\\var{b2}) \\pm \\sqrt{(\\var{b2})^2 -4 \\times (\\var{a2}) \\times (\\var{c2})}}{2 \\times (\\var{a2})} \\)</p>\n<p>Simplify this as much as possible (mainly the expression inside the square root and the denominator):</p>\n<p>\\( x= \\frac{\\simplify{-{b2}} \\pm \\sqrt{\\simplify{{b2}^2 -4*{a2}*{c2}}}}{\\simplify{2{a2}}} \\)</p>\n<p>Now, carry out this calculation twice, first using the plus (\\( + \\))&nbsp; and then using the minus (\\( - \\)).</p>\n<p>This will give you the two roots:&nbsp;&nbsp;&nbsp; \\( x = \\var{y1} \\)&nbsp; and &nbsp; \\( x = \\var{y2} \\)</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p>c)</p>\n<p>\\( \\simplify{ {a3}x^2 + {b3}x + {c3} } =0 \\)</p>\n<p>The co-efficients of this equation can simply be read from the equation, taking care to take account of positive and negative values:</p>\n<p>\\( a = \\)&nbsp; \\( \\var{a3}\\) &nbsp; &nbsp; \\( b = \\)&nbsp; \\( \\var{b3}\\) &nbsp; and&nbsp; \\( c = \\)&nbsp; \\( \\var{c3}\\)</p>\n<p>We now write the equation substituting these values in the correct places and, again, being careful with any negative values:</p>\n<p>\\( x= \\frac{-b \\pm \\sqrt{b^2 -4ac}}{2a} \\)&nbsp;&nbsp;&nbsp;&nbsp; becomes&nbsp;&nbsp;&nbsp;&nbsp; \\( x= \\frac{-(\\var{b3}) \\pm \\sqrt{(\\var{b3})^2 -4 \\times (\\var{a3}) \\times (\\var{c3})}}{2 \\times (\\var{a3})} \\)</p>\n<p>Simplify this as much as possible (mainly the expression inside the square root and the denominator):</p>\n<p>\\( x= \\frac{\\simplify{-{b3}} \\pm \\sqrt{\\simplify{{b3}^2 -4*{a3}*{c3}}}}{\\simplify{2{a3}}} \\)</p>\n<p>Now, carry out this calculation twice, first using the plus (\\( + \\))&nbsp; and then using the minus (\\( - \\)).</p>\n<p>This will give you the two roots:&nbsp;&nbsp;&nbsp; \\( x = \\var{z1} \\)&nbsp; and &nbsp; \\( x = \\var{z2} \\)</p>\n<p></p>\n<p></p>\n<p></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Question A", "definition": 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"adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>For the following equations, first identify the co-efficients, then use the formula to calculate the roots:</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{ {a1}x^2 + {b1}x + {c1} } =0 \\)</p>\n<p>The co-efficients of this equation are:</p>\n<p>\\( a = \\)&nbsp; [[0]] &nbsp; \\( b = \\)&nbsp; [[1]] &nbsp; and&nbsp; \\( c = \\)&nbsp; [[2]]</p>\n<p>&nbsp;</p>\n<p>Now use the formula to find the roots:</p>\n<p>\\( x_1 = \\)&nbsp; [[3]]</p>\n<p>\\( x_2 = \\)&nbsp; [[4]]</p>\n<p><sup><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><em>(If you are sure that your roots are correct but they are marked wrong, switch them around).</em></span></sup></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": "{a1}", "maxValue": "{a1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{b1}", "maxValue": "{b1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{c1}", "maxValue": "{c1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{x1}", "maxValue": "{x1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{x2}", "maxValue": "{x2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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{ {a2}x^2 + {b2}x + {c2} } =0 \\)</p>\n<p>The co-efficients of this equation are:</p>\n<p>\\( a = \\)&nbsp; [[0]] &nbsp; \\( b = \\)&nbsp; [[1]] &nbsp; and&nbsp; \\( c = \\)&nbsp; [[2]]</p>\n<p>&nbsp;</p>\n<p>Now use the formula to find the roots:</p>\n<p>\\( x_1 = \\)&nbsp; [[3]]</p>\n<p>\\( x_2 = \\)&nbsp; [[4]]</p>\n<p><sup><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><em>(If you are sure that your roots are correct but they are marked wrong, switch them around).</em></span></sup></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": "{a2}", "maxValue": "{a2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{b2}", "maxValue": "{b2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, 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"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{y2}", "maxValue": "{y2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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{ {a3}x^2 + {b3}x + {c3} } =0 \\)</p>\n<p>The co-efficients of this equation are:</p>\n<p>\\( a = \\)&nbsp; [[0]] &nbsp; \\( b = \\)&nbsp; [[1]] &nbsp; and&nbsp; \\( c = \\)&nbsp; [[2]]</p>\n<p>&nbsp;</p>\n<p>Now use the formula to find the roots:</p>\n<p>\\( x_1 = \\)&nbsp; [[3]]</p>\n<p>\\( x_2 = \\)&nbsp; [[4]]</p>\n<p><sup><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><em>(If you are sure that your roots are correct but they are marked wrong, switch them around).</em></span></sup></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], 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"nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{z1}", "maxValue": "{z1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{z2}", "maxValue": "{z2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Quadratic Equations: The Quadratic Formula 02", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "tags": [], "metadata": {"description": "<p>Solve quadratic equations (non-simple case, two real, discrete roots) using the formula. Some, random, non integer roots.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h4>If we have a quadratic equation of the form&nbsp; \\( ax^2 + bx +c = 0 \\),&nbsp; we can use the Quadratic Formula:</h4>\n<p>&nbsp;</p>\n<p style=\"text-align: center;\">\\( \\LARGE x= \\frac{-b \\pm \\sqrt{b^2 -4ac}}{2a} \\)</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p>to find the roots of the equation. (Note the plus-or-minus means there are two solutions, not just one.)</p>\n<p>It is also sometimes known as Shreedhara Acharya's formula, named after the 10<sup>th</sup> century Indian mathematician who first derived it.</p>", "advice": "<p>We are asked to find the roots of various quadratic equations using the Quadratic Formula. To enable us to do this, we are first asked to identify the co-efficients&nbsp; \\( a \\), \\( b \\) and \\( c \\) each time.</p>\n<p></p>\n<p>a)</p>\n<p>\\( \\simplify{ {a1}x^2 + {b1}x + {c1} } =0 \\)</p>\n<p>The co-efficients of this equation can simply be read from the equation, taking care to take account of positive and negative values:</p>\n<p>\\( a = \\)&nbsp; \\( \\var{a1}\\) &nbsp;&nbsp;&nbsp; \\( b = \\)&nbsp; \\( \\var{b1}\\) &nbsp; and&nbsp; \\( c = \\)&nbsp; \\( \\var{c1}\\)</p>\n<p>We now write the equation substituting these values in the correct places and, again, being careful with any negative values:</p>\n<p>\\( x= \\frac{-b \\pm \\sqrt{b^2 -4ac}}{2a} \\)&nbsp;&nbsp;&nbsp;&nbsp; becomes&nbsp;&nbsp;&nbsp;&nbsp; \\( x= \\frac{-(\\var{b1}) \\pm \\sqrt{(\\var{b1})^2 -4 \\times (\\var{a1}) \\times (\\var{c1})}}{2 \\times (\\var{a1})} \\)</p>\n<p>Simplify this as much as possible (mainly the expression inside the square root and the denominator):</p>\n<p>\\( x= \\frac{\\simplify{-{b1}} \\pm \\sqrt{\\simplify{{b1}^2 -4*{a1}*{c1}}}}{\\simplify{2{a1}}} \\)</p>\n<p>Now, carry out this calculation twice, first using the plus (\\( + \\))&nbsp; and then using the minus (\\( - \\)).</p>\n<p>This will give you the two roots:&nbsp;&nbsp;&nbsp; \\( x = \\var{x1} \\)&nbsp; and &nbsp; \\( x = \\var{x2} \\)</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p>b)</p>\n<p>\\( \\simplify{ {a2}x^2 + {b2}x + {c2} } =0 \\)</p>\n<p>The co-efficients of this equation can simply be read from the equation, taking care to take account of positive and negative values:</p>\n<p>\\( a = \\)&nbsp; \\( \\var{a2}\\) &nbsp; &nbsp; \\( b = \\)&nbsp; \\( \\var{b2}\\) &nbsp; and&nbsp; \\( c = \\)&nbsp; \\( \\var{c2}\\)</p>\n<p>We now write the equation substituting these values in the correct places and, again, being careful with any negative values:</p>\n<p>\\( x= \\frac{-b \\pm \\sqrt{b^2 -4ac}}{2a} \\)&nbsp;&nbsp;&nbsp;&nbsp; becomes&nbsp;&nbsp;&nbsp;&nbsp; \\( x= \\frac{-(\\var{b2}) \\pm \\sqrt{(\\var{b2})^2 -4 \\times (\\var{a2}) \\times (\\var{c2})}}{2 \\times (\\var{a2})} \\)</p>\n<p>Simplify this as much as possible (mainly the expression inside the square root and the denominator):</p>\n<p>\\( x= \\frac{\\simplify{-{b2}} \\pm \\sqrt{\\simplify{{b2}^2 -4*{a2}*{c2}}}}{\\simplify{2{a2}}} \\)</p>\n<p>Now, carry out this calculation twice, first using the plus (\\( + \\))&nbsp; and then using the minus (\\( - \\)).</p>\n<p>This will give you the two roots:&nbsp;&nbsp;&nbsp; \\( x = \\var{y1} \\)&nbsp; and &nbsp; \\( x = \\var{y2} \\)</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p>c)</p>\n<p>\\( \\simplify{ {a3}x^2 + {b3}x + {c3} } =0 \\)</p>\n<p>The co-efficients of this equation can simply be read from the equation, taking care to take account of positive and negative values:</p>\n<p>\\( a = \\)&nbsp; \\( \\var{a3}\\) &nbsp; &nbsp; \\( b = \\)&nbsp; \\( \\var{b3}\\) &nbsp; and&nbsp; \\( c = \\)&nbsp; \\( \\var{c3}\\)</p>\n<p>We now write the equation substituting these values in the correct places and, again, being careful with any negative values:</p>\n<p>\\( x= \\frac{-b \\pm \\sqrt{b^2 -4ac}}{2a} \\)&nbsp;&nbsp;&nbsp;&nbsp; becomes&nbsp;&nbsp;&nbsp;&nbsp; \\( x= \\frac{-(\\var{b3}) \\pm \\sqrt{(\\var{b3})^2 -4 \\times (\\var{a3}) \\times (\\var{c3})}}{2 \\times (\\var{a3})} \\)</p>\n<p>Simplify this as much as possible (mainly the expression inside the square root and the denominator):</p>\n<p>\\( x= \\frac{\\simplify{-{b3}} \\pm \\sqrt{\\simplify{{b3}^2 -4*{a3}*{c3}}}}{\\simplify{2{a3}}} \\)</p>\n<p>Now, carry out this calculation twice, first using the plus (\\( + \\))&nbsp; and then using the minus (\\( - \\)).</p>\n<p>This will give you the two roots:&nbsp;&nbsp;&nbsp; \\( x = \\var{z1} \\)&nbsp; and &nbsp; \\( x = \\var{z2} \\)</p>\n<p></p>\n<p></p>\n<p></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Question A", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Question A", "definition": "random(-9..9 #0.5 except 0 except 1 except {a1})", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Question A", "definition": "random(-9..9 except 0 except 1 except {a1} except {x1})", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Question A", "definition": "-{a1}*({x1}+{x2})", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Question A", "definition": "{a1}*{x1}*{x2}", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Question B", "definition": "random(2..9 except {a1})", "description": "", "templateType": "anything", "can_override": false}, "y1": {"name": "y1", "group": "Question B", "definition": "random(-9..9 #0.5 except 0 except 1 except {a2}) ", "description": "", "templateType": "anything", "can_override": false}, "y2": {"name": "y2", "group": "Question B", "definition": "random(-9..9 except 0 except 1 except {a2} except {y1})", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Question B", "definition": "-{a2}*({y1}+{y2})", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Question B", "definition": "{a2}*{y1}*{y2}", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Question C", "definition": "random(2..9 except {a1} except {a2})", "description": "", "templateType": "anything", "can_override": false}, "z1": {"name": "z1", "group": "Question C", "definition": "random(-9..9 except 0 except 1 except {a3})", "description": "", "templateType": "anything", "can_override": false}, "z2": {"name": "z2", "group": "Question C", "definition": "random(-9..9 #0.5 except 0 except 1 except {a3} except {z1})", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Question C", "definition": "-{a3}*({z1}+{z2})", "description": "", "templateType": "anything", "can_override": false}, "c3": {"name": "c3", "group": "Question C", "definition": "{a3}*{z1}*{z2}", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Question B", "variables": ["y1", "y2", "a2", "b2", "c2"]}, {"name": "Question C", "variables": ["z1", "z2", "a3", "b3", "c3"]}, {"name": "Question A", "variables": ["x1", "x2", "a1", "b1", "c1"]}], "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 equations, first identify the co-efficients, then use the formula to calculate the roots:</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{ {a1}x^2 + {b1}x + {c1} } =0 \\)</p>\n<p>The co-efficients of this equation are:</p>\n<p>\\( a = \\)&nbsp; [[0]] &nbsp; \\( b = \\)&nbsp; [[1]] &nbsp; and&nbsp; \\( c = \\)&nbsp; [[2]]</p>\n<p>&nbsp;</p>\n<p>Now use the formula to find the roots:</p>\n<p>\\( x_1 = \\)&nbsp; [[3]]</p>\n<p>\\( x_2 = \\)&nbsp; [[4]]</p>\n<p><sup><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><em>(If you are sure that your roots are correct but they are marked wrong, switch them around).</em></span></sup></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": "{a1}", "maxValue": "{a1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{b1}", "maxValue": "{b1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{c1}", "maxValue": "{c1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{x1}", "maxValue": "{x1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{x2}", "maxValue": "{x2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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{ {a2}x^2 + {b2}x + {c2} } =0 \\)</p>\n<p>The co-efficients of this equation are:</p>\n<p>\\( a = \\)&nbsp; [[0]] &nbsp; \\( b = \\)&nbsp; [[1]] &nbsp; and&nbsp; \\( c = \\)&nbsp; [[2]]</p>\n<p>&nbsp;</p>\n<p>Now use the formula to find the roots:</p>\n<p>\\( x_1 = \\)&nbsp; [[3]]</p>\n<p>\\( x_2 = \\)&nbsp; [[4]]</p>\n<p><sup><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><em>(If you are sure that your roots are correct but they are marked wrong, switch them around).</em></span></sup></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": "{a2}", "maxValue": "{a2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{b2}", "maxValue": "{b2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{c2}", "maxValue": "{c2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{y1}", "maxValue": "{y1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{y2}", "maxValue": "{y2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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{ {a3}x^2 + {b3}x + {c3} } =0 \\)</p>\n<p>The co-efficients of this equation are:</p>\n<p>\\( a = \\)&nbsp; [[0]] &nbsp; \\( b = \\)&nbsp; [[1]] &nbsp; and&nbsp; \\( c = \\)&nbsp; [[2]]</p>\n<p>&nbsp;</p>\n<p>Now use the formula to find the roots:</p>\n<p>\\( x_1 = \\)&nbsp; [[3]]</p>\n<p>\\( x_2 = \\)&nbsp; [[4]]</p>\n<p><sup><span class=\"prompt content-area\" localise-data-jme-context-description=\"part.prompt\"><em>(If you are sure that your roots are correct but they are marked wrong, switch them around).</em></span></sup></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": "{a3}", "maxValue": "{a3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{b3}", "maxValue": "{b3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{c3}", "maxValue": "{c3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{z1}", "maxValue": "{z1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{z2}", "maxValue": "{z2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}], "extensions": [], "custom_part_types": [], "resources": []}