// Numbas version: exam_results_page_options
{"name": "Quadratics: Solving a Quadratic Equation", "extensions": [], "custom_part_types": [{"source": {"pk": 2, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/2/edit"}, "name": "List of numbers", "short_name": "list-of-numbers", "description": "<p>The answer is a comma-separated list of numbers.</p>\n<p>The list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.</p>\n<p>You can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n  if(settings[\"correctAnswerFractions\"],\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n  ,\n    settings[\"correctAnswer\"]\n  ),\n  settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n  \"Enter a list of numbers separated by <code>{settings['separator']}</code>.\",\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )\n\nincluded:\nmap(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )\n\nall_included:\nall(included)\n\nno_extras:\nif(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )\n\ninterpreted_answer:\nif(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)\n\nmark:\nif(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)\n\nnotationStyles:\n[\"en\"]\n\nisSet:\nsettings[\"isSet\"]\n\nextra_numbers:\nfilter(not (x in expected_numbers),x,interpreted_answer)", "marking_notes": [{"name": "bits", "description": "<p>The separate items in the student's answer</p>", "definition": "let(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)"}, {"name": "expected_numbers", "description": "", "definition": "let(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)"}, {"name": "valid_numbers", "description": "<p>Is every number in the student's list valid?</p>", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )"}, {"name": "is_sorted", "description": "<p>Are the student's answers in ascending order?</p>", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )"}, {"name": "included", "description": "<p>Is each number in the expected answer present in the student's list the correct number of times?</p>", "definition": "map(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )"}, {"name": "all_included", "description": "<p>Has every number been included the right number of times?</p>", "definition": "all(included)"}, {"name": "no_extras", "description": "<p>True if the student's list doesn't contain any numbers that aren't in the expected answer.</p>", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "<p>Should the answer be considered as a set, so the number of times an element occurs doesn't matter?</p>", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "<p>Numbers included in the student's answer that are not in the expected list.</p>", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quadratics: Solving a Quadratic Equation", "tags": [], "metadata": {"description": "<p>Solving a quadratic equation of the form $ax^2+bx+c=0$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve the following quadratic equation:</p>\n<p>{question}</p>\n<p></p>", "advice": "<p>For a quadratic equation of the form \\[ ax^2+bx+c = 0,\\] we can use the quadratic formula to find solutions for $x$: \\[ x = \\frac{-b \\pm \\sqrt{b^2-4ac}}{2a}.\\]</p>\n<p>{advice}</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,a])", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-10..10 except [0,a,b])", "description": "", "templateType": "anything", "can_override": false}, "solx1": {"name": "solx1", "group": "Ungrouped variables", "definition": "if(f=1,(-b+sqrt(b^2-4*a*c))/(2*a),(-b1+sqrt(b1^2-4*a1*c1))/(2*a1))", "description": "", "templateType": "anything", "can_override": false}, "solx2": {"name": "solx2", "group": "Ungrouped variables", "definition": "if(f=1,(-b-sqrt(b^2-4*a*c))/(2*a),(-b1-sqrt(b1^2-4*a1*c1))/(2*a1))", "description": "", "templateType": "anything", "can_override": false}, "sol1": {"name": "sol1", "group": "Ungrouped variables", "definition": "precround(solx1,3)", "description": "", "templateType": "anything", "can_override": false}, "sol2": {"name": "sol2", "group": "Ungrouped variables", "definition": "precround(solx2,3)", "description": "", "templateType": "anything", "can_override": false}, "question": {"name": "question", "group": "Ungrouped variables", "definition": "if(f=1,'{q1}','{q2}')", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(1,2)", "description": "", "templateType": "anything", "can_override": false}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\simplify{{a}x^2+{b}x+{c}=0}\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "q2": {"name": "q2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\simplify{{m*n}x^2+{m*q+n*p}x+{p*q}=0}\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(1,2)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(1..2 except m)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(-5..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(-5..6 except[0,p])", "description": "", "templateType": "anything", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(f=2,'{solfac}','{solquad}')", "description": "", "templateType": "anything", "can_override": false}, "solfac": {"name": "solfac", "group": "Ungrouped variables", "definition": "\"<p>Note: In this case it is also possible to use factorisation to find solutions for $x$:</p>\\n<p>\\\\[ \\\\simplify{{m*n}x^2+{m*q+n*p}x+{p*q} = ({m}x+{p})({n}x+{q})}.\\\\]</p>\\n<p>However, we will give a worked solution using the quadratic formula, as this method holds for&nbsp;<em>all</em> quadratic equations of the form $ax^2+bx+c=0$.</p>\\n<p>For the equation \\\\[\\\\simplify{{a1}x^2+{b1}x+{c1}=0}, \\\\]</p>\\n<p>the values of the coefficients are \\\\[a=\\\\var{a1}, \\\\quad b=\\\\var{b1}, \\\\quad c=\\\\var{c1}.\\\\]</p>\\n<p>Substituting these values into the quadratic formula:</p>\\n<p>\\\\[ \\\\begin{split} x &amp;= \\\\frac{\\\\simplify{-{b1}} \\\\pm \\\\sqrt{(\\\\var{b1})^2 - (4 \\\\times \\\\var{a1} \\\\times \\\\var{c1})}}{2 \\\\times \\\\var{a1}} \\\\\\\\\\\\\\\\&amp;= \\\\frac{\\\\simplify{-{b1}} \\\\pm \\\\simplify[!collectNumbers]{sqrt({b1^2}-{4*a1*c1})}}{\\\\var{2*a1}} \\\\\\\\\\\\\\\\&amp;= \\\\frac{\\\\simplify{-{b1}} \\\\pm \\\\simplify[!collectNumbers]{sqrt({b1^2-4*a1*c1})}}{\\\\var{2*a1}} \\\\\\\\\\\\\\\\&amp;= \\\\frac{\\\\simplify{-{b1}} \\\\pm \\\\simplify{sqrt({b1^2-4*a1*c1})}}{\\\\var{2*a1}}. \\\\end{split} \\\\]</p>\\n<p>Note the $\\\\pm$ symbol in the quadratic formula. This means that there are $2$ solutions for $x$: one using $+$ and one using $-$.&nbsp;</p>\\n<p>Therefore,</p>\\n<p>\\\\[ \\\\begin{split} x_1 &amp;\\\\,= \\\\frac{\\\\simplify{-{b1}} + \\\\simplify{sqrt({b1^2-4*a1*c1})}}{\\\\var{2*a1}} \\\\qquad \\\\text{and} \\\\qquad&nbsp; x_2 &amp;\\\\,= \\\\frac{\\\\simplify{-{b1}} - \\\\simplify{sqrt({b1^2-4*a1*c1})}}{\\\\var{2*a1}}\\\\\\\\\\\\\\\\ &amp;\\\\,= \\\\var{sol1},\\\\quad &amp;\\\\,=\\\\var{sol2}. \\\\end{split}\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "solquad": {"name": "solquad", "group": "Ungrouped variables", "definition": "\"<p>For the equation&nbsp;\\\\[\\\\simplify{{a}x^2+{b}x+{c}=0}, \\\\]</p>\\n<p>the values of the coefficients are \\\\[a=\\\\var{a}, \\\\quad b=\\\\var{b}, \\\\quad c=\\\\var{c}.\\\\]</p>\\n<p>Substituting these values into the quadratic formula:</p>\\n<p>\\\\[ \\\\begin{split} x &amp;= \\\\frac{\\\\simplify{-{b}} \\\\pm \\\\sqrt{(\\\\var{b})^2 - (4 \\\\times \\\\var{a} \\\\times \\\\var{c})}}{2 \\\\times \\\\var{a}} \\\\\\\\\\\\\\\\&amp;= \\\\frac{\\\\simplify{-{b}} \\\\pm \\\\simplify[!collectNumbers]{sqrt({b^2}-{4*a*c})}}{\\\\var{2a}} \\\\\\\\\\\\\\\\&amp;= \\\\frac{\\\\simplify{-{b}} \\\\pm \\\\simplify[!collectNumbers]{sqrt({b^2-4*a*c})}}{\\\\var{2a}} \\\\\\\\\\\\\\\\&amp;= \\\\frac{\\\\simplify{-{b}} \\\\pm \\\\simplify{sqrt({b^2-4*a*c})}}{\\\\var{2a}}. \\\\end{split} \\\\]</p>\\n<p>Note the $\\\\pm$ symbol in the quadratic formula. This means that there are $2$ solutions for $x$: one using $+$ and one using $-$.&nbsp;</p>\\n<p>Therefore,</p>\\n<p>\\\\[ \\\\begin{split} x_1 &amp;\\\\,= \\\\frac{\\\\simplify{-{b}} + \\\\simplify{sqrt({b^2-4*a*c})}}{\\\\var{2a}} \\\\qquad \\\\text{and} \\\\qquad&nbsp; x_2 &amp;\\\\,= \\\\frac{\\\\simplify{-{b}} - \\\\simplify{sqrt({b^2-4*a*c})}}{\\\\var{2a}}\\\\\\\\\\\\\\\\ &amp;\\\\,= \\\\var{sol1},\\\\quad &amp;\\\\,=\\\\var{sol2}. \\\\end{split}\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "m*n", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "m*q+n*p", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "p*q", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "b^2-4a*c >0", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "a1", "m", "b1", "n", "c1", "p", "q", "solx1", "solx2", "sol1", "sol2", "question", "f", "q1", "q2", "advice", "solfac", "solquad"], "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": "<p>$x=$[[0]]&nbsp;</p>\n<p><em>Give&nbsp;your answers to 3 decimal places where necessary.</em></p>", "gaps": [{"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "[{sol1},{sol2}]", "allowFractions": false, "correctAnswerFractions": false, "isSet": false, "show_input_hint": true, "separator": ","}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}