// Numbas version: exam_results_page_options
{"name": "Inbbavathie's copy of Complete the square and find solutions (SP1)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["big", "sml", "bits"], "rulesets": {}, "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>"}, "parts": [{"gaps": [{"checkingaccuracy": 0.001, "musthave": {"strings": [")^2"], "partialCredit": 0, "message": "<p>It doesn't look like you've completed the square.</p>", "showStrings": false}, "answer": "(x+{bits[0]})^2-{bits[1]^2}", "checkingtype": "absdiff", "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 1, "expectedvariablenames": [], "notallowed": {"strings": ["x^2"], "partialCredit": 0, "message": "<p>It doesn't look like you've completed the square.</p>", "showStrings": false}, "showpreview": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "vsetrange": [0, 1]}], "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "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>", "variableReplacements": [], "marks": 0, "type": "gapfill"}, {"gaps": [{"checkingaccuracy": 0.001, "answer": "{-bits[0]-bits[1]}", "checkingtype": "absdiff", "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 1, "expectedvariablenames": [], "showpreview": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "vsetrange": [0, 1]}, {"checkingaccuracy": 0.001, "answer": "{-bits[0]+bits[1]}", "checkingtype": "absdiff", "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 1, "expectedvariablenames": [], "showpreview": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "vsetrange": [0, 1]}], "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "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>", "variableReplacements": [], "marks": 0, "type": "gapfill"}], "name": "Inbbavathie's copy of Complete the square and find solutions (SP1)", "variable_groups": [], "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>", "variables": {"sml": {"definition": "2*bits[0]", "group": "Ungrouped variables", "description": "<p>The coefficient of $x$ in the expanded quadratic.</p>", "name": "sml", "templateType": "anything"}, "big": {"definition": "bits[0]^2-bits[1]^2", "group": "Ungrouped variables", "description": "<p>The constant term in the expanded quadratic.</p>", "name": "big", "templateType": "anything"}, "bits": {"definition": "sort(shuffle(1..9)[0..2])", "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", "templateType": "anything"}}, "extensions": [], "preamble": {"js": "", "css": ""}, "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "contributors": [{"name": "Inbbavathie Ravi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1620/"}]}]}], "contributors": [{"name": "Inbbavathie Ravi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1620/"}]}