// Numbas version: finer_feedback_settings
{"name": "Katherine's copy of Complete the square and find solutions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve a quadratic equation by completing the square. The roots are not pretty!</p>"}, "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>", "parts": [{"scripts": {}, "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>", "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "showFeedbackIcon": true, "variableReplacements": [], "gaps": [{"checkingtype": "absdiff", "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "answer": "(x+{bits[0]})^2-{bits[1]^2}", "checkvariablenames": false, "vsetrange": [0, 1], "musthave": {"message": "<p>It doesn't look like you've completed the square.</p>", "partialCredit": 0, "strings": [")^2"], "showStrings": false}, "showCorrectAnswer": true, "notallowed": {"message": "<p>It doesn't look like you've completed the square.</p>", "partialCredit": 0, "strings": ["x^2"], "showStrings": false}, "scripts": {}, "marks": 1, "checkingaccuracy": 0.001, "type": "jme", "expectedvariablenames": [], "showpreview": true, "variableReplacements": [], "showFeedbackIcon": true}], "showCorrectAnswer": true}, {"scripts": {}, "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 style=\"text-align: left;\">Give the lowest solution first.</p>\n<p>$x_1=$&nbsp;[[0]]</p>\n<p>or</p>\n<p>$x_2=$&nbsp;[[1]]</p>", "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "showFeedbackIcon": true, "variableReplacements": [], "gaps": [{"checkingtype": "absdiff", "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "answer": "{-bits[0]-bits[1]}", "checkvariablenames": false, "vsetrange": [0, 1], "showCorrectAnswer": true, "scripts": {}, "marks": 1, "checkingaccuracy": 0.001, "type": "jme", "expectedvariablenames": [], "showpreview": true, "variableReplacements": [], "showFeedbackIcon": true}, {"checkingtype": "absdiff", "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "answer": "{-bits[0]+bits[1]}", "checkvariablenames": false, "vsetrange": [0, 1], "showCorrectAnswer": true, "scripts": {}, "marks": 1, "checkingaccuracy": 0.001, "type": "jme", "expectedvariablenames": [], "showpreview": true, "variableReplacements": [], "showFeedbackIcon": true}], "showCorrectAnswer": true}], "tags": [], "functions": {}, "variable_groups": [], "extensions": [], "ungrouped_variables": ["big", "sml", "bits"], "rulesets": {}, "variables": {"bits": {"group": "Ungrouped variables", "description": "<p>After completing the square, the expression will have the form $(x + \\mathrm{bits}[0])^2 - \\mathrm{bits}[1]^2$.</p>", "name": "bits", "definition": "sort(shuffle(1..9)[0..2])", "templateType": "anything"}, "big": {"group": "Ungrouped variables", "description": "<p>The constant term in the expanded quadratic.</p>", "name": "big", "definition": "bits[0]^2-bits[1]^2", "templateType": "anything"}, "sml": {"group": "Ungrouped variables", "description": "<p>The coefficient of $x$ in the expanded quadratic.</p>", "name": "sml", "definition": "2*bits[0]", "templateType": "anything"}}, "name": "Katherine's copy of Complete the square and find solutions", "preamble": {"js": "", "css": ""}, "statement": "", "type": "question", "contributors": [{"name": "Katherine Tomlinson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1652/"}], "resources": []}]}], "contributors": [{"name": "Katherine Tomlinson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1652/"}]}