// Numbas version: exam_results_page_options
{"name": "Quadratic expressions and equations", "metadata": {"description": "<p>Questions involving various techniques for rearranging and solving quadratic expressions and equations</p>", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Pre-calculus", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", ""], "questions": [{"name": "Factorising Quadratic Equations with $x^2$ Coefficients of 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}], "metadata": {"description": "<p>Factorise three quadratic equations of the form $x^2+bx+c$.</p>\n<p>The first has two negative roots, the second has one negative and one positive, and the third is the difference of two squares.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Factorise the following quadratic equations.</p>\n<p></p>", "variables": {"v1": {"name": "v1", "group": "Part A ", "definition": "random(1..10)", "templateType": "anything", "description": ""}, "v2": {"name": "v2", "group": "Part A ", "definition": "random(2..6 except v1)", "templateType": "anything", "description": ""}, "v4": {"name": "v4", "group": "Part A ", "definition": "random(1..10 except -v3)", "templateType": "anything", "description": ""}, "v5": {"name": "v5", "group": "Part A ", "definition": "random(2..10)", "templateType": "anything", "description": ""}, "v3": {"name": "v3", "group": "Part A ", "definition": "random(-8..-1)", "templateType": "anything", "description": ""}, "v6": {"name": "v6", "group": "Part A ", "definition": "-v5", "templateType": "anything", "description": ""}}, "tags": ["Factorisation", "factorisation", "factorising quadratic equations", "Factorising quadratic equations", "taxonomy"], "ungrouped_variables": [], "functions": {}, "preamble": {"js": "question.is_factorised = function(part,penalty) {\n  penalty = penalty || 0;\n  if(part.credit>0) {\n    // Parse the student's answer as a syntax tree\n    var studentTree = Numbas.jme.compile(part.studentAnswer,Numbas.jme.builtinScope);\n\n    // Create the pattern to match against \n    // we just want two sets of brackets, each containing two terms\n    // or one of the brackets might not have a constant term\n    // or for repeated roots, you might write (x+a)^2\n    var rule = Numbas.jme.compile('m_all(m_any(x,x+m_pm(m_number),x^m_number,(x+m_pm(m_number))^m_number))*m_nothing');\n\n    // Check the student's answer matches the pattern. \n    var m = Numbas.jme.display.matchTree(rule,studentTree,true);\n    // If not, take away marks\n    if(!m) {\n      part.multCredit(penalty,'Your answer is not fully factorised.');\n    }\n  }\n}", "css": ""}, "advice": "<p>Quadratic equations&nbsp;of the form</p>\n<p>\\[x^2+bx+c=0\\]</p>\n<p>can be factorised to&nbsp;create&nbsp;an equation of the form</p>\n<p>\\[(x+m)(x+n)=0\\text{.}\\]</p>\n<p>When we expand a factorised quadratic expression we obtain</p>\n<p>\\[(x+m)(x+n)=x^2+(m+n)x+(m \\times n)\\text{.}\\]</p>\n<p>To factorise an equation of the form $x^2+bx+c$, we need to find two numbers which add together to make $b$, and multiply together to make $c$.</p>\n<h4>a)</h4>\n<p>\\[\\simplify{x^2+{v1+v2}x+{v1*v2}=0}\\]</p>\n<p>We need to find two values that add together to make $\\var{v1+v2}$ and multiply together to make $\\var{v1*v2}$.</p>\n<p>\\[\\begin{align}<br/>\\var{v1} \\times \\var{v2}&amp;=\\var{v1*v2}\\\\<br/>\\var{v1}+\\var{v2}&amp;=\\var{v1+v2}\\\\<br/>\\end{align} \\]</p>\n<p>So the factorised form of the equation is</p>\n<p>\\[\\simplify{(x+{v1})(x+{v2})}=0\\text{.}\\]</p>\n<p></p>\n<h4>b)</h4>\n<p>We can begin factorising by finding factors of $\\var{v3*v4}$ that add together to give $\\var{v3+v4}$.</p>\n<p>\\[\\begin{align}<br/>\\var{v3} \\times \\var{v4}&amp;=\\var{v3*v4}\\\\<br/>\\var{v3}+\\var{v4}&amp;=\\var{v3+v4}\\\\<br/>\\end{align} \\]</p>\n<p>So the factorised form of the equation is</p>\n<p>\\[\\simplify{(x+{v3})(x+{v4})}=0\\text{.}\\]</p>\n<h4>c)</h4>\n<p>When factorising the quadratic expression</p>\n<p>\\[\\simplify{x^2+{v5*v6}=0}\\]</p>\n<p>we need to find two values that add together to make $0$ and multiply together to make $\\var{v5*v6}$.</p>\n<p>\\begin{align}<br/>\\var{v5} \\times \\var{v6}&amp; = \\var{v5*v6}\\\\<br/>\\simplify[]{ {v5} + {v6}} &amp;= 0 \\\\<br/>\\end{align}</p>\n<p>So the factorised form of the equation is</p>\n<p>\\[\\simplify{(x+{v5})(x+{v6})}=0\\text{.}\\]</p>", "type": "question", "variable_groups": [{"variables": ["v1", "v2", "v3", "v4", "v5", "v6"], "name": "Part A "}], "parts": [{"scripts": {}, "variableReplacements": [], "customName": "", "useCustomName": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"mustmatchpattern": {"pattern": "(`+-x^$n`? + `+- $n)`* * $z", "message": "Your answer is not fully factorised.", "partialCredit": 0, "nameToCompare": ""}, "variableReplacementStrategy": "originalfirst", "unitTests": [], "checkingAccuracy": 0.001, "scripts": {}, "failureRate": 1, "checkVariableNames": false, "marks": 1, "valuegenerators": [{"name": "x", "value": ""}], "variableReplacements": [], "useCustomName": false, "showPreview": true, "customName": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "answer": "(x+{v1})(x+{v2})", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "vsetRangePoints": 5}], "type": "gapfill", "marks": 0, "customMarkingAlgorithm": "", "prompt": "<p>$\\simplify{x^2+{v1+v2}x+{v1*v2}=0}$</p>\n<p>[[0]]&nbsp;$=0$</p>\n<p><em></em></p>", "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "variableReplacements": [], "customName": "", "useCustomName": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"mustmatchpattern": {"pattern": "(`+-x^$n`? + `+- $n)`* * $z", "message": "Your answer is not fully factorised.", "partialCredit": 0, "nameToCompare": ""}, "variableReplacementStrategy": "originalfirst", "unitTests": [], "checkingAccuracy": 0.001, "scripts": {}, "failureRate": 1, "checkVariableNames": false, "marks": 1, "valuegenerators": [{"name": "x", "value": ""}], "variableReplacements": [], "useCustomName": false, "showPreview": true, "customName": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "answer": "(x+{v3})(x+{v4})", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "vsetRangePoints": 5}], "type": "gapfill", "marks": 0, "customMarkingAlgorithm": "", "prompt": "<p>$\\simplify{x^2+{v3+v4}x+{v3*v4}}=0$</p>\n<p>[[0]] $=0$</p>\n<p><em></em></p>", "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "variableReplacements": [], "customName": "", "useCustomName": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"mustmatchpattern": {"pattern": "(`+-x^$n`? + `+- $n)`* * $z", "message": "<div>Your answer is not fully factorised.</div>", "partialCredit": 0, "nameToCompare": ""}, "variableReplacementStrategy": "originalfirst", "unitTests": [], "checkingAccuracy": 0.001, "scripts": {}, "failureRate": 1, "checkVariableNames": false, "marks": 1, "valuegenerators": [{"name": "x", "value": ""}], "variableReplacements": [], "useCustomName": false, "showPreview": true, "customName": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "answer": "(x+{v5})(x+{v6})", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "vsetRangePoints": 5}], "type": "gapfill", "marks": 0, "customMarkingAlgorithm": "", "prompt": "<p><em></em>$\\simplify{x^2+{v5*v6}}=0$</p>\n<p>[[0]] $=0$</p>", "variableReplacementStrategy": "originalfirst"}], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}}, {"name": "Factorising Quadratic Equations with $x^2$ Coefficients Greater than 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}], "advice": "<h4>a)</h4>\n<p>As this question involves a number&nbsp;greater than $1$ before the $x^2$ value&nbsp;it has a factorised form $(ax+b)(cx+d)$.</p>\n<p>To find $a$ and $c$, we need to consider the factors of $\\var{a*c}$.</p>\n<p>We are already given that one of them is $\\var{a}$, so we know that the other one must be $\\var{c}$.</p>\n<p>This means our factorised equation must take the form</p>\n<p>\\[(\\var{a}x+b)(\\var{c}x+d)=0\\text{.}\\]</p>\n<p>This expands to</p>\n<p>\\[ \\simplify{ {a*c}x^2 + ({a}*d+{c}*b)x + a*b} \\]</p>\n<p>So we must find two numbers which add together to make $\\var{a*d+b*c}$, and multiply together to make $\\var{b*d}$.</p>\n<p>Therefore $b$ and $d$ must satisfy</p>\n<p>\\begin{align}<br/>b \\times d &amp;=\\var{b*d}\\\\<br/>\\simplify{{a}d+{c}b} &amp;= \\var{a*d+b*c}\\text{.}<br/>\\end{align}</p>\n<p>$b = \\var{b}$ and $d = \\var{d}$ satisfy these equations:</p>\n<p>\\begin{align}<br/>\\var{b} \\times \\var{d} &amp;=\\var{b*d}\\\\<br/>\\simplify[]{ {a}*{d} + {b}*{c} } &amp;= \\var{a*d+b*c}<br/>\\end{align}</p>\n<p>So the factorised form of the equation is&nbsp;</p>\n<p>\\[ \\simplify{({a}x+{b})({c}x+{d}) = 0} \\text{.}\\]</p>\n<h4>b)</h4>\n<p>$\\simplify{({a}x+{b})({c}x+{d}) = 0}$ when either $\\var{a}x+\\var{b} = 0$ or $\\var{c}x+ \\var{d} = 0$.</p>\n<p>So the roots of the equation are $\\var[fractionnumbers]{-b/a}$ and $\\var[fractionnumbers]{-d/c}$.</p>\n<p></p>", "statement": "", "variables": {"b": {"templateType": "anything", "name": "b", "description": "<p>$b$ in $(ax+b)(cx+d)$</p>", "group": "last q", "definition": "random(-5..5 except 0)"}, "c": {"templateType": "anything", "name": "c", "description": "<p><strong></strong>$c$ in $(ax+b)(cx+d)$</p>", "group": "last q", "definition": "random(2..8 except a)"}, "a": {"templateType": "anything", "name": "a", "description": "<p>$a$ in $(ax+b)(cx+d)$</p>", "group": "last q", "definition": "random(2..3)"}, "roots": {"templateType": "anything", "name": "roots", "description": "<p>The roots of the equation</p>", "group": "last q", "definition": "sort([-b/a,-d/c])"}, "d": {"templateType": "anything", "name": "d", "description": "<p>$d$ in $(ax+b)(cx+d)$</p>", "group": "last q", "definition": "random(-8..8 except 0)"}}, "tags": ["coefficient of x^2 greater than 1", "Factorisation", "factorisation", "factorising", "factorising quadratic equations", "Factorising quadratic equations", "factorising quadratic equations with x^2 coefficients greater than 1", "taxonomy"], "ungrouped_variables": [], "functions": {}, "rulesets": {}, "metadata": {"description": "<p>Factorise a quadratic equation where the coefficient of the $x^2$ term is greater than 1 and then write down the roots of the equation</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "variable_groups": [{"variables": ["a", "b", "c", "d", "roots"], "name": "last q"}], "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "b", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "b", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "c", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "c", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "d", "correctAnswerFraction": false, "allowFractions": false, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "d", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "marks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "<p>Factorise the equation</p>\n<p>$\\simplify{{a*c}x^2+{a*d+b*c}x+{b*d}=0}\\text{.}$</p>\n<p>$(\\var{a}x+\\phantom{.}$[[0]]$) ($[[1]]$x+\\phantom{.}$[[2]]$)\\; = 0$</p>", "variableReplacements": [], "showFeedbackIcon": true}, {"scripts": {}, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "roots[0]", "correctAnswerFraction": true, "allowFractions": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "roots[0]", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}, {"notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "variableReplacements": [], "mustBeReducedPC": 0, "minValue": "roots[1]", "correctAnswerFraction": true, "allowFractions": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "maxValue": "roots[1]", "showCorrectAnswer": true, "type": "numberentry", "marks": 1, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "marks": 0, "variableReplacementStrategy": "originalfirst", "prompt": "<p></p>\n<p>Write down the roots of the equation above.</p>\n<p>Input your answer as $x_1$ and $x_2$, where $x_1&lt;x_2$.</p>\n<p>$x_1=$ [[0]]</p>\n<p>$x_2=$&nbsp;[[1]]</p>", "variableReplacements": [], "showFeedbackIcon": true}]}, {"name": "Algebra: Solving quadratics by factorising", "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}, "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}], "tags": [], "metadata": {"description": "<p>Some quadratics are to be solved by factorising</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Solve the quadratic equations by factorising.&nbsp;If there is more than solution, enter them all separated by a comma.</p>\n<p>-----------------------------------</p>", "advice": "<p>See 5.1 and 5.2 for how to use factorising to solve quadratics.</p>\n<p>See 3.3 and 3.5 for factorising.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"d": {"name": "d", "group": "Ungrouped variables", "definition": "map(num[j][3],j,0..4)", "description": "", "templateType": "anything"}, "solmin": {"name": "solmin", "group": "Ungrouped variables", "definition": "vector([min(-d[0]/c[0],-b[0]/a[0])]+[min(-d[1]/c[1],-b[1]/a[1])]+[min(-d[2]/c[2],-b[2]/a[2])]+[min(-d[3]/c[3],-b[3]/a[3])]+[min(-d[4]/c[4],-b[4]/a[4])])", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "map(num[j][1],j,0..4)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "map(num[j][2],j,0..4)", "description": "", "templateType": "anything"}, "solmax": {"name": "solmax", "group": "Ungrouped variables", "definition": "vector([max(-d[0]/c[0],-b[0]/a[0])]+[max(-d[1]/c[1],-b[1]/a[1])]+[max(-d[2]/c[2],-b[2]/a[2])]+[max(-d[3]/c[3],-b[3]/a[3])]+[max(-d[4]/c[4],-b[4]/a[4])])", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "map(num[j][0],j,0..4)", "description": "", "templateType": "anything"}, "num": {"name": "num", "group": "Ungrouped variables", "definition": "[[1,random(6..7),1,random(2..5)]] +\nshuffle([\n  [1,random(-4..-6),1,random(-3..3 except 0)],\n  [1,random(-4..-6),random(2..3),random(-1..1 except 0)],\n  [1,random(1,3,5),4,random(1,3,5)*random([1,-1])],\n  [1,random(-4..-6),3,random([-1,1,-2,2])]\n  ])", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["num", "a", "b", "c", "d", "solmax", "solmin"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Factorise $\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+ {b[0]*d[0]}}$.&nbsp;[[0]]</p>\n<p>Hence solve $\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+ {b[0]*d[0]}}=0$.</p>\n<p>[[1]]</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({a[0]}x+{b[0]})({c[0]}x+{d[0]})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "musthave": {"strings": [")("], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["x^2", "x*x", "x x", "x(", "x*("], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "settings": {"correctAnswer": "[solmin[0],solmax[0]]", "allowFractions": true, "correctAnswerFractions": true}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Factorise $\\simplify{{a[1]*c[1]}x^2+{a[1]*d[1]+b[1]*c[1]}x+ {b[1]*d[1]}}$.&nbsp;[[0]]</p>\n<p>Hence solve $\\simplify{{a[1]*c[1]}x^2+{a[1]*d[1]+b[1]*c[1]}x+ {b[1]*d[1]}}=0$.</p>\n<p>[[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({a[1]}x+{b[1]})({c[1]}x+{d[1]})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "musthave": {"strings": [")("], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["x^2", "x*x", "x x", "x(", "x*("], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "settings": {"correctAnswer": "[solmin[1],solmax[1]]", "allowFractions": true, "correctAnswerFractions": true}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Factorise $\\simplify{{a[2]*c[2]}x^2+{a[2]*d[2]+b[2]*c[2]}x+ {b[2]*d[2]}}$.&nbsp;[[0]]</p>\n<p>Hence solve $\\simplify{{a[2]*c[2]}x^2+{a[2]*d[2]+b[2]*c[2]}x+ {b[2]*d[2]}}=0$.</p>\n<p>[[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({a[2]}x+{b[2]})({c[2]}x+{d[2]})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "musthave": {"strings": [")("], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["x^2", "x*x", "x x", "x(", "x*("], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "settings": {"correctAnswer": "[solmin[2],solmax[2]]", "allowFractions": true, "correctAnswerFractions": true}}], "sortAnswers": false}]}, {"name": "Completing the Square", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "tags": [], "metadata": {"description": "<p>Write the expression $ax^2+bx+c$ in completed square form $a(x+p)^2+k$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Rewrite the expression \\[\\simplify{{a}x^2+{b}x+{c}}\\] in completed square form&nbsp;$a(x+k)^2+p$ for suitable numbers $a, k$ and $p$.</p>\n<p><strong></strong></p>\n<p><strong></strong>Your answer must be input exactly in this form.</p>", "advice": "<p>(a)</p>\n<p>Given the quadratic $q(x)=\\simplify{{a}x^2+{b}x+ {c}}$, one way of completing the square is to simply expand the term $a(x+k)^2+p$ and equate coefficients to the quadratic term.&nbsp;</p>\n<p>Expanding&nbsp; $a(x+k)^2+p$ gives $ax^2 + 2akx + ak^2 + p$.&nbsp;</p>\n<p>Equating coefficients gives:</p>\n<p>$x^2:\\quad \\var{a}=a$ or $a=\\var{a}$.</p>\n<p>$x:\\quad \\var{b}= 2ak \\implies k =&nbsp; \\simplify{{b}/(2{a})}$</p>\n<p>constant: $\\var{c}=ak^2+p \\implies p=\\var{c}-ak^2=\\simplify[all, fractionNumbers]{{c-b^2/(4a)}}$.</p>\n<p>Now, put these together to obtain: $\\qquad \\simplify[all, fractionNumbers]{{a}(x+{b}/(2{a}))^2+{c-b^2/(4a)}}$.</p>\n<hr/>\n<p>(b)</p>\n<p>Completing the square for this function gives:$\\qquad \\simplify[all, fractionNumbers]{{a}(x+{b}/(2{a}))^2+{c-b^2/(4a)}}$.</p>\n<p>The function attains its turning point when $\\quad \\simplify{{a}(x+{b}/(2{a}))^2}=0$&nbsp;</p>\n<p>So the $x$ coordinate of the turning point is $\\simplify{-{b}/(2{a})}$.</p>\n<p>The $y$ coordinate is $k=\\simplify[all, fractionNumbers]{{c-b^2/(4a)}}$.</p>\n<p>Hence the turning point is $\\quad (\\simplify{-{b}/(2{a})},\\simplify[all, fractionNumbers]{{c-b^2/(4a)}})$</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5..7 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-20..20)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "(a)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify{{a}x^2+{b}x+ {c}}&nbsp;= \\phantom{{}}$ [[0]].</p>", "stepsPenalty": "2", "steps": [{"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>Given the quadratic $q(x)=\\simplify{{a}x^2+{b}x+ {c}}$, one way of completing the square is to simply expand the term $a(x+k)^2+p$ and equate coefficients to the quadratic term.&nbsp;</p>\n<p>Expanding&nbsp; $a(x+k)^2+p$ gives $ax^2 + 2akx + ak^2 + p$.&nbsp;</p>\n<p>Equating coefficients gives:</p>\n<p>$x^2:\\quad \\var{a}=a$ or $a=\\var{a}$.</p>\n<p>$x:\\quad \\var{b}= 2ak \\implies k =&nbsp; \\simplify{{b}/(2{a})}$</p>\n<p>constant: $\\var{c}=ak^2+p \\implies p=\\var{c}-ak^2=\\simplify[all, fractionNumbers]{{c-b^2/(4a)}}$.</p>\n<p>Now, put these together to obtain: $\\qquad \\simplify[all, fractionNumbers]{{a}(x+{b}/(2{a}))^2+{c-b^2/(4a)}}$.</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": "{a}(x+{b}/(2{a}))^2+{c}-{b}^2/(4{a})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.0001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(", ")", "^"], "showStrings": false, "partialCredit": 0, "message": "<p>please input in the form $(x+a)^2+b$</p>"}, "notallowed": {"strings": ["x^2", "x*x", "x x", "x(", "x*("], "showStrings": false, "partialCredit": 0, "message": "<p>Input your answer in the form $(x+a)^2+b$.</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "(b)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Hence, state the coordinates of the turning point for $q(x)=\\simplify{{a}x^2+{b}x+ {c}}$:</p>\n<p>Turning point:$\\quad \\phantom{}$[[0]].</p>", "stepsPenalty": 1, "steps": [{"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>Completing the square for this function gives:$\\qquad \\simplify[all, fractionNumbers]{{a}(x+{b}/(2{a}))^2+{c-b^2/(4a)}}$.</p>\n<p>The function attains its turning point when $\\quad \\simplify{{a}(x+{b}/(2{a}))^2}=0$&nbsp;</p>\n<p>So the $x$ coordinate of the turning point is $\\simplify{-{b}/(2{a})}$.</p>\n<p>The $y$ coordinate is $k=\\simplify[all, fractionNumbers]{{c-b^2/(4a)}}$.</p>\n<p>Hence the turning point is $\\quad (\\simplify{-{b}/(2{a})},\\simplify[all, fractionNumbers]{{c-b^2/(4a)}})$</p>"}], "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([-{b}/(2{a}),{c}-{b}^2/(4{a})])", "correctAnswerFractions": true, "numRows": 1, "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Complete the square and find solutions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}], "variable_groups": [], "variables": {"bits": {"templateType": "anything", "description": "<p>After completing the square, the expression will have the form $(x + \\mathrm{bits}[0])^2 - \\mathrm{bits}[1]^2$.</p>", "definition": "sort(shuffle(1..9)[0..2])", "name": "bits", "group": "Ungrouped variables"}, "big": {"templateType": "anything", "description": "<p>The constant term in the expanded quadratic.</p>", "definition": "bits[0]^2-bits[1]^2", "name": "big", "group": "Ungrouped variables"}, "sml": {"templateType": "anything", "description": "<p>The coefficient of $x$ in the expanded quadratic.</p>", "definition": "2*bits[0]", "name": "sml", "group": "Ungrouped variables"}}, "parts": [{"type": "gapfill", "showCorrectAnswer": true, "unitTests": [], "prompt": "<p style=\"text-align: left;\"><em></em>Write the following expression in the form $a(x+b)^2-c$.</p>\n<p style=\"text-align: left;\">$\\simplify {x^2+{sml}x+{big}} = $ [[0]]<em></em></p>", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "scripts": {}, "customMarkingAlgorithm": "", "gaps": [{"showCorrectAnswer": true, "checkVariableNames": false, "useCustomName": false, "unitTests": [], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "musthave": {"partialCredit": 0, "showStrings": false, "strings": [")^2"], "message": "<p>It doesn't look like you've completed the square.</p>"}, "variableReplacementStrategy": "originalfirst", "failureRate": 1, "notallowed": {"partialCredit": 0, "showStrings": false, "strings": ["x^2"], "message": "<p>It doesn't look like you've completed the square.</p>"}, "vsetRange": [0, 1], "variableReplacements": [], "customMarkingAlgorithm": "", "vsetRangePoints": 5, "adaptiveMarkingPenalty": 0, "customName": "", "type": "jme", "checkingAccuracy": 0.001, "answer": "(x+{bits[0]})^2-{bits[1]^2}", "valuegenerators": [{"value": "", "name": "x"}], "checkingType": "absdiff", "scripts": {}, "showPreview": true, "marks": 1}], "marks": 0, "variableReplacements": [], "useCustomName": false, "adaptiveMarkingPenalty": 0, "customName": ""}, {"type": "gapfill", "showCorrectAnswer": true, "unitTests": [], "prompt": "<p style=\"text-align: left;\">Now solve the quadratic equation</p>\n<p style=\"text-align: left;\">\\[ \\simplify {x^2+{sml}x+{big}} = 0\\text{.} \\]</p>\n<p>$x_1=$&nbsp;[[0]]</p>\n<p>or</p>\n<p>$x_2=$&nbsp;[[1]]</p>", "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "sortAnswers": true, "scripts": {}, "customMarkingAlgorithm": "", "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "showCorrectAnswer": true, "unitTests": [], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "minValue": "{-bits[0]-bits[1]}", "maxValue": "{-bits[0]-bits[1]}", "showFractionHint": true, "variableReplacements": [], "customMarkingAlgorithm": "", "adaptiveMarkingPenalty": 0, "customName": "", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "useCustomName": false, "correctAnswerStyle": "plain", "scripts": {}, "marks": 1}, {"correctAnswerFraction": false, "mustBeReduced": false, "showCorrectAnswer": true, "unitTests": [], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "minValue": "{-bits[0]+bits[1]}", "maxValue": "{-bits[0]+bits[1]}", "showFractionHint": true, "variableReplacements": [], "customMarkingAlgorithm": "", "adaptiveMarkingPenalty": 0, "customName": "", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "useCustomName": false, "correctAnswerStyle": "plain", "scripts": {}, "marks": 1}], "marks": 0, "variableReplacements": [], "useCustomName": false, "adaptiveMarkingPenalty": 0, "customName": ""}], "advice": "<p>Completing the square works by noticing that</p>\n<p>\\[ (x+a)^2 = x^2 + 2ax + a^2 \\]</p>\n<p>So when we see an expression of the form $x^2 + 2ax$, we can rewrite it as $(x+a)^2-a^2$.</p>\n<h4>a)</h4>\n<p>Rewrite $x^2+\\var{sml}x$ as $\\simplify[basic]{ (x+{sml/2})^2 - {sml/2}^2}$.</p>\n<p>\\begin{align}<br/>\\simplify[basic]{x^2+{sml}x+{big}} &amp;= \\simplify[basic]{(x+{sml/2})^2-{(sml/2)}^2+{big}} \\\\<br/>&amp;= \\simplify[basic]{(x+{sml/2})^2+{-(sml/2)^2+big}} \\text{.}<br/>\\end{align}</p>\n<h4>b)</h4>\n<p>We showed above that</p>\n<p>\\[&nbsp;\\simplify[basic]{x^2+{sml}x+{big}} = 0 \\]</p>\n<p>is equivalent to</p>\n<p>\\[&nbsp;\\simplify[basic]{(x+{bits[0]})^2-{bits[1]^2}} = 0 \\text{.} \\]</p>\n<p>We can then rearrange this equation to solve for $x$.</p>\n<p>\\begin{align}<br/>\\simplify{(x+{bits[0]})^2-{(bits[1])^2} } &amp;= 0 \\\\<br/>(x+\\var{bits[0]})^2 &amp;= \\var{bits[1]^2} \\\\<br/>x+\\var{bits[0]} &amp;= \\pm&nbsp;\\var{bits[1]} \\\\<br/>x &amp;= -\\var{bits[0]} \\pm \\var{bits[1]}&nbsp;\\\\[2em]<br/><br/>x_1 &amp;= \\var{-bits[0]-bits[1]} \\text{,}\\\\<br/>x_2 &amp;= \\var{-bits[0]+bits[1]} \\text{.}<br/>\\end{align}</p>", "tags": ["taxonomy"], "preamble": {"css": "", "js": ""}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["big", "sml", "bits"], "statement": "<p>We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a&nbsp;square plus another term - this is called \"completing the square\".</p>\n<p>This can be useful when it isn't obvious how to fully factorise a quadratic equation.</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve a quadratic equation by completing the square. The roots are not pretty!</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}}, {"name": "Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}], "metadata": {"description": "<p>Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "ungrouped_variables": ["a1", "a2", "a3", "a4", "b1", "b2", "b3", "b4", "x1", "p1", "p2", "x2", "a", "m"], "rulesets": {}, "advice": "<p>The quadratic formula is&nbsp;</p>\n<p>\\[x={\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\\]</p>\n<h4>a)</h4>\n<p>From the equation, we can read off values for $a$, $b$ and $c$:</p>\n<p>\\[\\begin{align}<br/>a&amp;=1\\text{,}\\\\<br/>b&amp;=\\var{a+m}\\text{,}\\\\<br/>c&amp;=\\var{a*m} \\text{.}<br/>\\end{align}\\]</p>\n<p>Substituting&nbsp;these values into the quadratic formula,</p>\n<p>\\[x = \\frac {-\\var{a+m}\\pm\\sqrt{\\var{a+m}^2-4\\times \\var{a*m}}}{2}\\text{.}\\]</p>\n<p>Note the $\\pm$ symbol in the formula. This means there are two solutions: one using $+$, the other using $-$.</p>\n<p>The two solutions are</p>\n<p>\\[\\begin{align}<br/>x_1&amp;=\\var{m}\\text{,}\\\\<br/>x_2&amp;=\\var{a}\\text{.}<br/>\\end{align}\\]</p>\n<h4>b)</h4>\n<p><em></em>Note that the right-hand side of the given equation is not zero. We need to&nbsp;rewrite it in the form $ax^2+bx+c=0$:</p>\n<p>\\[\\begin{align}<br/>\\simplify{{a1}x^2+{a2}x+{a3}}&amp;=\\var{a4}\\\\<br/>\\simplify{{a1}x^2+{a2}x+{a3-a4}}&amp;=0\\text{.}<br/>\\end{align}\\]</p>\n<p>Then we can read off values for $a$, $b$ and $c$:</p>\n<p>\\[\\begin{align}<br/>a&amp;=\\var{a1}\\\\<br/>b&amp;=\\var{a2}\\\\<br/>c&amp;=\\var{a3-a4} \\text{.}<br/>\\end{align}\\]</p>\n<p>We can now substitute these values into the quadratic formula:</p>\n<p>\\[x = {\\frac {-\\var{a2}\\pm\\sqrt{\\var{a2}^2-4\\times \\var{a1}\\times \\var{a3-a4}}}{2\\times\\var{a1}}}\\text{.}\\]</p>\n<p>So the two solutions are</p>\n<p>\\[\\begin{align}<br/>x_1&amp;=\\var{dpformat(x1,2)}\\\\<br/>x_2&amp;=\\var{dpformat(x2,2)}\\text{.}<br/>\\end{align}\\]</p>\n<h4>c)</h4>\n<p>We first rearrange our equation&nbsp;into the form $ax^2+bx+c=0$:</p>\n<p>\\[\\begin{align}<br/>\\simplify{{b1}x^2+{b2}x+{b3}}&amp;=0=\\var{b4}x\\\\<br/>\\simplify{{b1}x^2+{b2-b4}x+{b3}}&amp;=0\\text{.}<br/>\\end{align}\\]</p>\n<p>We can then read off the&nbsp;values for $a, b$ and $c$, which are</p>\n<p>\\[\\begin{align}<br/>a&amp;=\\var{b1}\\text{,}\\\\<br/>b&amp;=\\var{b2-b4}\\text{,}\\\\<br/>c&amp;=\\var{b3}\\text{.}<br/>\\end{align}\\]</p>\n<p>Substituting&nbsp;these values into the quadratic formula,</p>\n<p>\\[x = {\\frac {-\\var{b2-b4}\\pm\\sqrt{\\var{b2-b4}^2-4\\times \\var{b1}\\times \\var{b3}}}{2\\times\\var{b1}}},\\]</p>\n<p>we obtain solutions</p>\n<p>\\[\\begin{align}<br/>x_1&amp;=\\var{dpformat(p1,2)}\\text{,}\\\\<br/>x_2&amp;=\\var{dpformat(p2,2)}\\text{.}<br/>\\end{align}\\]</p>", "variable_groups": [{"name": "part 2", "variables": ["b", "c"]}], "statement": "<p>When quadratic equations can't be factorised, or&nbsp;if equations are difficult to factorise (perhaps if the coefficients are large), we need to use the quadratic formula to solve the equations.</p>\n<p>Use the quadratic formula to calculate values for $x$ in these equations. Input the possible values as $x_1$ and $x_2$, where $x_1&lt;x_2$.</p>", "parts": [{"scripts": {}, "variableReplacements": [], "type": "gapfill", "prompt": "<p>$\\simplify{x^2+{a+m}x+{a*m}=0}$</p>\n<p>$x_1=$&nbsp;[[0]]</p>\n<p>$x_2=$&nbsp;[[1]]<em><br/></em></p>", "stepsPenalty": 0, "steps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>An equation of the form</p>\n<p>\\[ax^2+bx+c=0\\text{,}\\]</p>\n<p></p>\n<p>can be solved using the quadratic formula</p>\n<p>\\[x={\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\\]</p>\n<p></p>"}], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "correctAnswerFraction": false, "precision": "2", "precisionPartialCredit": 0, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "precisionType": "dp", "minValue": "-a", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "-a", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "strictPrecision": false, "showPrecisionHint": true}, {"allowFractions": false, "correctAnswerFraction": false, "precision": "2", "precisionPartialCredit": 0, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "precisionType": "dp", "minValue": "-m", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "-m", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "strictPrecision": false, "showPrecisionHint": true}], "showFeedbackIcon": true, "marks": 0, "showCorrectAnswer": true}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "prompt": "<p></p>\n<p>$\\simplify{{a1}x^2+{a2}x+{a3}={a4}}$</p>\n<p>$x_1=$&nbsp;[[0]]</p>\n<p>$x_2=$&nbsp;[[1]]</p>\n<p><em></em></p>", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "correctAnswerFraction": false, "precision": "2", "precisionPartialCredit": 0, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "precisionType": "dp", "minValue": "x1", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "x1", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "strictPrecision": false, "showPrecisionHint": true}, {"allowFractions": false, "correctAnswerFraction": false, "precision": "2", "precisionPartialCredit": 0, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "precisionType": "dp", "minValue": "x2", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "x2", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "strictPrecision": false, "showPrecisionHint": true}], "showFeedbackIcon": true, "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "prompt": "<p>$\\simplify{{b1}x^2+{b2}x+{b3}={b4}x}$</p>\n<p>$x_1=$&nbsp;[[0]]</p>\n<p>$x_2=$&nbsp;[[1]]</p>", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "correctAnswerFraction": false, "precision": "2", "precisionPartialCredit": 0, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "precisionType": "dp", "minValue": "p1", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "p1", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "strictPrecision": false, "showPrecisionHint": true}, {"allowFractions": false, "correctAnswerFraction": false, "precision": "2", "precisionPartialCredit": 0, "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showFeedbackIcon": true, "precisionType": "dp", "minValue": "p2", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "p2", "mustBeReduced": false, "marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "strictPrecision": false, "showPrecisionHint": true}], "showFeedbackIcon": true, "marks": 0}], "tags": ["Quadratic formula", "quadratic formula", "solve quadratic equations", "taxonomy", "Using the quadratic formula", "using the quadratic formula"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"a3": {"description": "", "definition": "random(-30..4 except 0)", "group": "Ungrouped variables", "name": "a3", "templateType": "anything"}, "a": {"description": "", "definition": "random(15..20)", "group": "Ungrouped variables", "name": "a", "templateType": "anything"}, "a1": {"description": "", "definition": "random(2..9)", "group": "Ungrouped variables", "name": "a1", "templateType": "anything"}, "m": {"description": "", "definition": "random(9..a-1)", "group": "Ungrouped variables", "name": "m", "templateType": "anything"}, "b1": {"description": "", "definition": "random(2..10 except a1)", "group": "Ungrouped variables", "name": "b1", "templateType": "anything"}, "c": {"description": "", "definition": "random(-10..4 except 0)^2", "group": "part 2", "name": "c", "templateType": "anything"}, "b4": {"description": "", "definition": "random(-5..5)", "group": "Ungrouped variables", "name": "b4", "templateType": "anything"}, "p2": {"description": "", "definition": "(-(b2-b4)+((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "group": "Ungrouped variables", "name": "p2", "templateType": "anything"}, "b3": {"description": "", "definition": "random(-30..6 except a3)", "group": "Ungrouped variables", "name": "b3", "templateType": "anything"}, "a4": {"description": "", "definition": "random(1..15)", "group": "Ungrouped variables", "name": "a4", "templateType": "anything"}, "x1": {"description": "", "definition": "(-(a2)-((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)", "group": "Ungrouped variables", "name": "x1", "templateType": "anything"}, "p1": {"description": "", "definition": "(-(b2-b4)-((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "group": "Ungrouped variables", "name": "p1", "templateType": "anything"}, "b": {"description": "", "definition": "2c^0.5+a1^2", "group": "part 2", "name": "b", "templateType": "anything"}, "b2": {"description": "", "definition": "random(20..35 except a2)", "group": "Ungrouped variables", "name": "b2", "templateType": "anything"}, "a2": {"description": "", "definition": "random(9..30 except a1)", "group": "Ungrouped variables", "name": "a2", "templateType": "anything"}, "x2": {"description": "", "definition": "(-(a2)+((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)", "group": "Ungrouped variables", "name": "x2", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Use the quadratic formula to solve an equation in terms of an unknown variable", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}], "variable_groups": [{"variables": ["b", "c", "n2", "b_2", "c_2", "b_3"], "name": "part 2"}], "preamble": {"css": "", "js": ""}, "type": "question", "parts": [{"marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "gaps": [{"checkingtype": "absdiff", "type": "jme", "vsetrangepoints": 5, "vsetrange": [0, 1], "showpreview": true, "answer": "({-b_3/2-sqrt(b_3^2-4*c_2)/2})k", "scripts": {}, "showFeedbackIcon": true, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 1, "checkingaccuracy": 0.001, "variableReplacements": [], "expectedvariablenames": []}, {"checkingtype": "absdiff", "type": "jme", "vsetrangepoints": 5, "vsetrange": [0, 1], "showpreview": true, "answer": "({-b_3/2+sqrt(b_3^2-4*c_2)/2})k", "scripts": {}, "showFeedbackIcon": true, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 1, "checkingaccuracy": 0.001, "variableReplacements": [], "expectedvariablenames": []}], "stepsPenalty": 0, "showFeedbackIcon": true, "prompt": "<p>Solve the equation $\\simplify {x^2+{b_3}*k*x+{c_2}k^2=0}$. Give your answer in terms of $k$. Assuming $k$ is positive, enter the lowest root first.</p>\n<p>$x_1=$&nbsp;[[0]]</p>\n<p>$x_2=$&nbsp;[[1]]</p>", "steps": [{"variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "<p>The quadratic formula is</p>\n<p>\\[{\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\\]</p>"}]}], "advice": "<p>The quadratic formula is</p>\n<p>\\[{\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\\]</p>\n<p>We can list our values for $a, b$ and $c$.</p>\n<p>\\[\\begin{align}<br/>a&amp;=1\\\\<br/>b&amp;=\\var{b_3}k\\\\<br/>c&amp;=\\var{c_2}k^2<br/>\\end{align}\\]</p>\n<p>Then by substituting them into the quadratic formula, we obtain</p>\n<p>\\[x=\\frac {-\\var{b_3}k\\pm\\sqrt{\\var{b_3}^2k^2-4\\times \\var{c_2}k^2}}{2}\\]</p>\n<p>We can then simplify this equation to</p>\n<p>\\[\\begin{align}<br/>x&amp;=\\frac {-\\var{b_3}k\\pm k\\sqrt{\\var{b_3}^2-\\var{4c_2}}}{2}\\\\<br/>\\end{align}\\]<br/>\\[\\begin{align}<br/>&amp;=k\\left(\\frac{-\\var{b_3}}{2}\\pm&nbsp;\\frac {\\sqrt{\\var{b_3}^2-\\var{4c_2}}}{2}\\right)\\\\<br/>\\end{align}\\]<br/>\\[\\begin{align}<br/>&amp;=k\\left(-\\frac{\\var{b_3}}{2} \\pm\\frac{\\sqrt{\\var{(b_3^2-4c_2)}}}{2}\\right)\\\\<br/>\\end{align}\\]</p>\n<p>This means our possible values for $x$ in terms of $k$ are,</p>\n<p>\\begin{align}<br/>x_1 &amp;= \\left( \\simplify[all,!noleadingminus,!collectnumbers,!simplifyfractions]{-{b_3}/2 - {sqrt(b_3^2-4c_2)}/2} \\right) k = \\var[fractionnumbers]{-b_3/2 - sqrt(b_3^2-4*c_2)/2}k \\\\<br/>x_2 &amp;= \\left( \\simplify[all,!noleadingminus,!collectnumbers,!simplifyfractions]{-{b_3}/2 +&nbsp;{sqrt(b_3^2-4c_2)}/2} \\right) k = \\var[fractionnumbers]{-b_3/2 + sqrt(b_3^2-4*c_2)/2}k<br/>\\end{align}</p>", "tags": ["Quadratic equations", "Quadratic Equations", "quadratic equations", "quadratic formula", "Quadratic formula", "solving quadratic equations", "Solving quadratic equations", "Solving quadratic equations using the quadratic formula", "taxonomy", "unknown variable", "Unknown variable"], "variables": {"b": {"templateType": "anything", "description": "", "definition": "2c^0.5+a1^2", "name": "b", "group": "part 2"}, "m": {"templateType": "anything", "description": "", "definition": "random(9..a)", "name": "m", "group": "Ungrouped variables"}, "n2": {"templateType": "anything", "description": "", "definition": "random(1..10)^2", "name": "n2", "group": "part 2"}, "b_2": {"templateType": "anything", "description": "", "definition": "random(n2^0.5..11 except n2^0.5)^2 ", "name": "b_2", "group": "part 2"}, "b1": {"templateType": "anything", "description": "", "definition": "random(1..10 except a1)", "name": "b1", "group": "Ungrouped variables"}, "p1": {"templateType": "anything", "description": "", "definition": "(-(b2-b4)-((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "name": "p1", "group": "Ungrouped variables"}, "a4": {"templateType": "anything", "description": "", "definition": "random(1..15)", "name": "a4", "group": "Ungrouped variables"}, "a1": {"templateType": "anything", "description": "", "definition": "random(1..9)", "name": "a1", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "description": "", "definition": "random(-10..4 except 0)^2", "name": "c", "group": "part 2"}, "x1": {"templateType": "anything", "description": "", "definition": "(-(a2)-((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)", "name": "x1", "group": "Ungrouped variables"}, "b_3": {"templateType": "anything", "description": "", "definition": "b_2^0.5", "name": "b_3", "group": "part 2"}, "c_2": {"templateType": "anything", "description": "", "definition": "(b_2-n2)/4", "name": "c_2", "group": "part 2"}, "a3": {"templateType": "anything", "description": "", "definition": "random(-30..4 except 0)", "name": "a3", "group": "Ungrouped variables"}, "p2": {"templateType": "anything", "description": "", "definition": "(-(b2-b4)+((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "name": "p2", "group": "Ungrouped variables"}, "b2": {"templateType": "anything", "description": "", "definition": "random(20..35 except a2)", "name": "b2", "group": "Ungrouped variables"}, "a": {"templateType": "anything", "description": "", "definition": "random(15..20)", "name": "a", "group": "Ungrouped variables"}, "a2": {"templateType": "anything", "description": "", "definition": "random(9..30 except a1)", "name": "a2", "group": "Ungrouped variables"}, "x2": {"templateType": "anything", "description": "", "definition": "(-(a2)+((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)", "name": "x2", "group": "Ungrouped variables"}, "b4": {"templateType": "anything", "description": "", "definition": "random(-5..5)", "name": "b4", "group": "Ungrouped variables"}, "b3": {"templateType": "anything", "description": "", "definition": "random(-30..6 except a3)", "name": "b3", "group": "Ungrouped variables"}}, "rulesets": {}, "functions": {}, "ungrouped_variables": ["a1", "a2", "a3", "a4", "b1", "b2", "b3", "b4", "x1", "p1", "p2", "x2", "a", "m"], "statement": "<p>The coefficients of the following equation involve the unknown value $k$. We can use the quadratic formula to find expressions for the values of $x$ in terms of $k$.</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Factorise a quadratic expression of the form $x^2+akx+bk^2$ for $x$, in terms of $k$. $a$ and $b$ are constants.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}}]}], "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": []}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "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": []}