// Numbas version: finer_feedback_settings
{"name": "10.j. Solving (gx+a)/b=c/(fx+d)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "10.j. Solving (gx+a)/b=c/(fx+d)", "tags": [], "metadata": {"description": "<p>quadratics with some set up rearranging needed</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve the following equation for $x$.</p>\n<p>\\[\\simplify{({g}x+{a})/{b}}=\\simplify{{c}/({f}x+{d})}\\]</p>", "advice": "<p>First multiply through by $\\var{b}$ and $\\simplify{{f}x+{d}}$ and re-arrange to get&nbsp;</p>\n<p>\\[\\simplify{{x2Coff}x^2+{xCoff}x+{Const}=0}.\\]</p>\n<p>To solve a quadratic equation of the form \\[ ax^2+bx+c=0\\] by factorisation, we want to factorise the equation into the form \\[(mx+p)(nx+q)=0,\\] where $mn=a$, $np+mq=b$, and $p \\times q = c$.&nbsp;</p>\n<p>The equation \\[\\simplify{{x2Coff}x^2+{xCoff}x+{Const}=0}\\]</p>\n<p>can be factorised to \\[\\simplify{({a1}x+{Fact1})({a2}x+{Fact2})=0}.\\] This equation is satisfied when either \\[\\simplify{{a1}x+{Fact1}=0} \\quad \\text{or} \\quad \\simplify{{a2}x+{Fact2}=0}, \\] which implies the solutions to this quadratic equation are \\[ \\simplify[fractionNumbers]{x={-x1}} \\quad \\text{and} \\quad \\simplify[fractionNumbers]{x={-x2}} .\\]</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Quadratics\" target=\"_blank\" rel=\"noopener\">Use this link to find resources to help you revise how to solve quadratic equations.</a></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"x1": {"name": "x1", "group": "Ungrouped variables", "definition": "Fact1/a1", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "Fact2/a2", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(separate(-50..50,x->x|(Const-a*d))[0])", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "-(Const-a*d)/b", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "VarChoice[2]", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "VarChoice[3]", "description": "", "templateType": "anything", "can_override": false}, "xCoff": {"name": "xCoff", "group": "Ungrouped variables", "definition": "a1*Fact2+a2*Fact1", "description": "<p>Coefficient of x when written in form ax^2+bx+c</p>", "templateType": "anything", "can_override": false}, "Const": {"name": "Const", "group": "Ungrouped variables", "definition": "Fact1*Fact2", "description": "<p>Constant term when written in form ax^2+bx+c</p>", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "VarChoice[1]", "description": "", "templateType": "anything", "can_override": false}, "x2Coff": {"name": "x2Coff", "group": "Ungrouped variables", "definition": "a1*a2", "description": "<p>Coefficient of x^2 when written in form ax^2+bx+c</p>", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "VarChoice[0]", "description": "", "templateType": "anything", "can_override": false}, "Fact1": {"name": "Fact1", "group": "Ungrouped variables", "definition": "random(-12..12)", "description": "<p>polynomial factorises as (a1x+Fact1)(a2x+Fact2)</p>", "templateType": "anything", "can_override": false}, "Fact2": {"name": "Fact2", "group": "Ungrouped variables", "definition": "random(-12..12)", "description": "<p>polynomial factorises as (a1x+Fact1)(a2x+Fact2)</p>", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "VarList": {"name": "VarList", "group": "Ungrouped variables", "definition": "[x,x2Coff/x,y,(xCoff-x2Coff/x*y)/x] for: x of:separate(-25..25,x->x|(a1*a2))[0] for: y of:-50..50", "description": "", "templateType": "anything", "can_override": false}, "VarListAllInt": {"name": "VarListAllInt", "group": "Ungrouped variables", "definition": "separate(VarList, x -> int(x[3])=x[3] && x[2]*x[3]<>Const)[0]", "description": "", "templateType": "anything", "can_override": false}, "VarChoice": {"name": "VarChoice", "group": "Ungrouped variables", "definition": "random(VarListAllInt)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "a*d<>0 and b*c<>0 and x1<>a and x2<>d and x2<>a and x1<>d", "maxRuns": "100"}, "ungrouped_variables": ["x1", "x2", "a", "b", "c", "d", "g", "f", "xCoff", "Const", "x2Coff", "Fact1", "Fact2", "a1", "a2", "VarList", "VarListAllInt", "VarChoice"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "student_set (The set of numbers that the student gave):\n  set(x for: x of: interpreted_answers where: x<>nothing)\n\nexpected_set (The set of roots of the equation):\n  set([-x1,-x2])\n\nnum_correct (The number of values in the student's set which are correct):\n  len(intersection(student_set, expected_set))\n\nmark:\n  if(num_correct=0,\n    incorrect(\"Your answer is incorrect.\")\n  , \n    if(len(expected_set)=1, // there is one root\n      if(len(student_set)=1, // student gave one correct root\n        correct()\n      , add_credit(0.5, \"One of your answers is not a root.\") // student gave one correct and one incorrect)\n      )\n    , // there are two 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