// Numbas version: exam_results_page_options {"name": "Quadratic simultaneous equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quadratic simultaneous equations", "tags": [], "metadata": {"description": "Solving 1 linear and 1 quadratic simultaneous equations", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Solve the simultaneous equations:

\n

$x+2y=\\var{x+2y}$

\n

$x^2+y^2=\\var{x^2+y^2}$

", "advice": "

Rearranging the first equation gives $x=\\var{x+2y}-2y$

\n

Substituting this expression for $x$ into our second equation gives $(\\var{x+2y}-2y)^2+y^2=\\var{x^2+y^2}$

\n

Expanding brackets gives $\\simplify[]{{(x+2y)^2}-{4*(x+2y)}}$ $y+4y^2+y^2=\\var{x^2+y^2}$

\n

Rearranging gives $\\simplify[]{5y^2-{4*(x+2y)}y-{x^2+y^2-(x+2y)^2}}=0$

\n

Solving by factorising or using the quadratic formula gives $y=\\frac{\\var{4a} \\pm \\sqrt{\\var{16a^2+20(b-a^2)}}}{10}$

\n

Hence $y = \\var{y1}$ or $\\var{y2}$

\n

and the corresponding values of $x$ are $\\var{x1}$ and $\\var{x2}$ respectively (found by substituting each value of $y$ into $x+2y=\\var{x+2y}$)

", "rulesets": {}, "extensions": [], "variables": {"x": {"name": "x", "group": "Ungrouped variables", "definition": "random(-8..8 except 0)", "description": "", "templateType": "anything"}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "random(-8..8 except 0 except x)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "x+2y", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "x^2+y^2", "description": "", "templateType": "anything"}, "y1": {"name": "y1", "group": "Ungrouped variables", "definition": "(2a-sqrt(5b-a^2))/5", "description": "", "templateType": "anything"}, "y2": {"name": "y2", "group": "Ungrouped variables", "definition": "((2a+sqrt(5b-a^2)))/5", "description": "", "templateType": "anything"}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "a-2y1", "description": "", "templateType": "anything"}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "a-2y2", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x", "y", "a", "b", "y1", "y2", "x1", "x2"], "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": "

Lower value of $y=$[[0]]

\n

Corresponding value of $x=$[[1]]

\n

\n

Greater value of $y=$[[2]]

\n

Corresponding value of $x=$[[3]]

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "y1", "maxValue": "y1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "x1", "maxValue": "x1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "y2", "maxValue": "y2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "x2", "maxValue": "x2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "contributors": [{"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}