// Numbas version: exam_results_page_options
{"name": "Ecuaci\u00f3n de cuarto grado reducible a Cuadr\u00e1tica ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "variable_groups": [{"name": "Unnamed group", "variables": []}], "preamble": {"js": "", "css": ""}, "statement": "<p>Resolver el siguiente ecuaci&oacute;n:</p>\n<p>\\[\\begin{eqnarray*}\\simplify[std]{x^4-{a^2+b^2}x^2+{(a*b)^2}}&amp;=&amp;{0}\\end{eqnarray*}\\]</p>", "ungrouped_variables": ["a", "b", "c"], "name": "Ecuaci\u00f3n de cuarto grado reducible a Cuadr\u00e1tica ", "functions": {}, "metadata": {"description": "<p>Shows how to define variables to stop degenerate examples.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {"std": ["All", "fractionnumbers"]}, "parts": [{"scripts": {}, "gaps": [{"scripts": {}, "checkvariablenames": false, "checkingtype": "absdiff", "showFeedbackIcon": true, "variableReplacements": [], "marks": 1, "expectedvariablenames": [], "showCorrectAnswer": true, "checkingaccuracy": 0.001, "answersimplification": "Std", "showpreview": true, "notallowed": {"partialCredit": 0, "strings": ["."], "showStrings": false, "message": "<p>Input your answer as a fraction and not a decimal.</p>"}, "vsetrange": [0, 1], "type": "jme", "variableReplacementStrategy": "originalfirst", "answer": "{-b}", "vsetrangepoints": 5}, {"scripts": {}, "checkvariablenames": false, "checkingtype": "absdiff", "showFeedbackIcon": true, "variableReplacements": [], "marks": 1, "expectedvariablenames": [], "showCorrectAnswer": true, "checkingaccuracy": 0.001, "answersimplification": "Std", "showpreview": true, "notallowed": {"partialCredit": 0, "strings": ["."], "showStrings": false, "message": "<p>Input your answer as a fraction and not a decimal.</p>"}, "vsetrange": [0, 1], "type": "jme", "variableReplacementStrategy": "originalfirst", "answer": "{-a}", "vsetrangepoints": 5}, {"scripts": {}, "checkvariablenames": false, "checkingtype": "absdiff", "showFeedbackIcon": true, "variableReplacements": [], "marks": 1, "expectedvariablenames": [], "showCorrectAnswer": true, "checkingaccuracy": 0.001, "answersimplification": "Std", "showpreview": true, "notallowed": {"partialCredit": 0, "strings": ["."], "showStrings": false, "message": "<p>Input your answer as a fraction and not a decimal.</p>"}, "vsetrange": [0, 1], "type": "jme", "variableReplacementStrategy": "originalfirst", "answer": "{a}", "vsetrangepoints": 5}, {"scripts": {}, "checkvariablenames": false, "checkingtype": "absdiff", "showFeedbackIcon": true, "variableReplacements": [], "marks": 1, "expectedvariablenames": [], "showCorrectAnswer": true, "checkingaccuracy": 0.001, "answersimplification": "Std", "showpreview": true, "notallowed": {"partialCredit": 0, "strings": ["."], "showStrings": false, "message": "<p>Input your answer as a fraction and not a decimal.</p>"}, "vsetrange": [0, 1], "type": "jme", "variableReplacementStrategy": "originalfirst", "answer": "{b}", "vsetrangepoints": 5}], "showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p><strong>Observaci&oacute;n:</strong>&nbsp;Para ingresar su respuesta, h&aacute;galo en el siguiente orden:</p>\n<p>$x_1&lt;x_2&lt;x_3&lt;x_4$</p>\n<p>$x_1=$ [[0]]&nbsp;;&nbsp;$x_2=$ [[1]]&nbsp;;&nbsp;$x_3=$[[2]]&nbsp;;&nbsp;$x_4=$[[3]]&nbsp;&nbsp;</p>\n<p></p>\n<p></p>"}], "extensions": [], "tags": [], "advice": "<p>Utilicemos la&nbsp;sustituci&oacute;n $\\,x^2=t\\,$ en la ecuaci&oacute;n:</p>\n<p>\\[\\begin{eqnarray*}\\simplify[std]{x^4-{a^2+b^2}x^2+{(a*b)^2}}&amp;=&amp;{0}\\end{eqnarray*}\\]</p>\n<p>Nos queda:</p>\n<p>\\[\\begin{eqnarray*}\\simplify[std]{t^2-{a^2+b^2}t+{(a*b)^2}}&amp;=&amp;{0}\\end{eqnarray*}\\]</p>\n<p>Esta ecuaci&oacute;n la podemos resolver factorizando.</p>\n<p>\\begin{align}<br/>(\\simplify{t-{a^2}})(\\simplify{t-{b^2}})&amp;=\\,{0}<br/>\\end{align}</p>\n<p>\\[\\simplify{(t-{a^2}={0})} \\,\\,\\,\\,;\\,\\,\\,&nbsp;\\simplify{(t-{b^2}={0})} \\]</p>\n<p>\\[\\simplify{(t={a^2})} \\,\\,\\,\\,;\\,\\,\\,&nbsp;\\simplify{(t={b^2})} \\]</p>\n<p>Ahora nos devolvemos con la sustituci&oacute;n&nbsp;$\\,x^2=t\\,$, de modo de encontrar los valores de la variable $x$:</p>\n<p>\\[\\simplify{(x^2={a^2})} \\,\\,\\,\\,;\\,\\,\\,&nbsp;\\simplify{(x^2={b^2})} \\]</p>\n<p>Estas ecuaciones las podemos resolver nuevamente factorizando:</p>\n<p>\\[\\simplify{(x^2-{a^2}={0})} \\,\\,\\,\\,;\\,\\,\\,&nbsp;\\simplify{(x^2-{b^2}={0})} \\]</p>\n<p>\\begin{align}<br/>(\\simplify{x-{a}})(\\simplify{x+{a}})&amp;=\\,{0}\\,\\,\\,\\,;\\,\\,\\,(\\simplify{x-{b}})(\\simplify{x+{b}})=\\,{0}<br/>\\end{align}</p>\n<p>\\begin{align}<br/>\\simplify{(x={a})} \\,\\,\\,\\,;\\,\\,\\,&nbsp;\\simplify{x={-a}}\\,\\,\\,\\,;\\,\\,\\,\\simplify{(x={b})} \\,\\,\\,\\,;\\,\\,\\,&nbsp;\\simplify{x={-b}}<br/>\\end{align}</p>\n<p></p>\n<p>Finalmente ordenamos las soluciones de menor a mayor como lo pide el enunciado:</p>\n<p>\\begin{align}<br/>\\simplify{(x_1={-b})} \\,\\,\\,\\,;\\,\\,\\,&nbsp;\\simplify{x_2={-a}}\\,\\,\\,\\,;\\,\\,\\,\\simplify{(x_3={a})} \\,\\,\\,\\,;\\,\\,\\,&nbsp;\\simplify{x_4={b}}<br/>\\end{align}</p>\n<p></p>", "variables": {"a": {"name": "a", "definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "c": {"name": "c", "definition": "1", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b": {"name": "b", "definition": "a+random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}}, "type": "question", "contributors": [{"name": "Patricio Ramirez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/191/"}]}]}], "contributors": [{"name": "Patricio Ramirez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/191/"}]}