// Numbas version: exam_results_page_options
{"name": "cormac's copy of Solve quadratic inequalities", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "name": "cormac's copy of Solve quadratic inequalities", "statement": "<p>Solve the following quadratic inequalities by firstly factorising $f(x)$ and then solving for $x$ when $f(x)=0$. It may be helpful to sketch each&nbsp;quadratic.</p>", "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>This question takes the student through variety of examples of quadratic inequalities by asking them for the range(s) for which $x$ meets the inequality.</p>"}, "rulesets": {}, "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a", "b", "c", "d", "g", "f"], "variables": {"d": {"templateType": "anything", "name": "d", "group": "Ungrouped variables", "description": "", "definition": "random(2..4)"}, "b": {"templateType": "anything", "name": "b", "group": "Ungrouped variables", "description": "", "definition": "random(1..5)"}, "g": {"templateType": "anything", "name": "g", "group": "Ungrouped variables", "description": "", "definition": "random(1..4)"}, "f": {"templateType": "anything", "name": "f", "group": "Ungrouped variables", "description": "", "definition": "random(5..9)"}, "c": {"templateType": "anything", "name": "c", "group": "Ungrouped variables", "description": "", "definition": "random(3..6)"}, "a": {"templateType": "anything", "name": "a", "group": "Ungrouped variables", "description": "", "definition": "random(1..5)"}}, "advice": "<h3>a)</h3>\n<p>This example&nbsp;is best illustrated by the graph of $\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}$ below. By finding the roots of the equation we can find the $x$ coordinates where the line crosses the $x$ axis and then we can use a sketch or visualise the graph to work out the set of values for $x$ where $f(x)&gt;0$.\\[\\\\[0.1em]\\]</p>\n<p style=\"text-align: center;\">{geogebra_applet('Hk5dTptY',[[\"a\",a],[\"b\",b]])}</p>\n<p>\\[ <br/>\\begin{align}<br/>\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}&amp;&gt;0\\\\<br/>\\simplify{(x+{a})(x-{b})}&amp;=0\\text{.}<br/>\\end{align}<br/>\\]</p>\n<p>Therefore</p>\n<p>\\[ \\simplify{x&lt;{-a}}\\text{ or&nbsp;}\\simplify{x&gt;{b}}\\text{.}\\]</p>\n<h4>b)</h4>\n<p>We use the same method in this example but this time we use&nbsp;our graph to visualise where $g(x)&lt;0$.\\[\\\\[0.1em]\\]</p>\n<p style=\"text-align: center;\">{geogebra_applet('PUFStTNa',[[\"c\",c],[\"d\",d]])}</p>\n<p>\\[ <br/>\\begin{align}<br/>\\simplify{f(x)=x^2+({c}-{d})x-{c}{d}}&amp;&lt;0\\\\<br/>\\simplify{(x+{c})(x-{d})}&amp;=0\\text{.}<br/>\\end{align}<br/>\\]</p>\n<p>Therefore</p>\n<p>\\[ \\simplify{x&gt;{-c}}\\text{ and }\\simplify{x&lt;{d}}\\text{.}\\]</p>\n<h4>c)</h4>\n<p style=\"text-align: left;\">The notable difference with solving this equation is the requirement to rearrange the inequality&nbsp;before factorisation.\\[\\\\[0.1em]\\]</p>\n<p style=\"text-align: center;\">{geogebra_applet('CZsSCdH6',[[\"e\",g],[\"f\",f]])}</p>\n<p>\\[ <br/>\\begin{align}<br/>\\simplify{({g}+{f})x-{g}{f}}&amp;&gt;x^2\\\\<br/>\\simplify{x^2-({g}+{f})x+{g}{f}}&amp;&lt;0\\\\<br/>\\simplify{(x-{g})(x-{f})}&amp;=0\\text{.}<br/>\\end{align}<br/>\\]</p>\n<p>Therefore</p>\n<p>\\[ \\simplify{x&gt;{g}}\\text{ and }\\simplify{x&lt;{f}}\\text{.}\\]</p>\n<p>Alternatively, we can plot the graph of $x^2$ against $\\simplify{({g}+{f})x-{g}{f}}$ to visualise the same result.</p>\n<p style=\"text-align: center;\">{geogebra_applet('Ez697SNE',[[\"g\",g],[\"f\",f]])}</p>\n<p style=\"text-align: left;\">From this graph we can see that the values of $x$ where&nbsp;$\\simplify{({g}+{f})x-{g}{f}}&gt;x^2$ are the same as the values of $x$ where $\\simplify{x^2-({g}+{f})x+{g}{f}}&lt;0$.</p>", "extensions": ["geogebra"], "functions": {}, "parts": [{"unitTests": [], "variableReplacements": [], "showCorrectAnswer": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "gaps": [{"unitTests": [], "expectedVariableNames": [], "variableReplacements": [], "showCorrectAnswer": true, "customMarkingAlgorithm": "", "checkingAccuracy": 0.001, "checkVariableNames": false, "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "vsetRange": [0, 1], "marks": 1, "showPreview": true, "scripts": {}, "type": "jme", "checkingType": "absdiff", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "answer": "(x-{g})(x-{f})"}, {"unitTests": [], "expectedVariableNames": [], "variableReplacements": [], "showCorrectAnswer": true, "customMarkingAlgorithm": "", "checkingAccuracy": 0.001, "checkVariableNames": false, "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "vsetRange": [0, 1], "marks": 1, "showPreview": true, "scripts": {}, "type": "jme", "checkingType": "absdiff", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "answer": "{g}"}, {"unitTests": [], "expectedVariableNames": [], "variableReplacements": [], "showCorrectAnswer": true, "customMarkingAlgorithm": "", "checkingAccuracy": 0.001, "checkVariableNames": false, "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "vsetRange": [0, 1], "marks": 1, "showPreview": true, "scripts": {}, "type": "jme", "checkingType": "absdiff", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "answer": "{f}"}, {"unitTests": [], "displayColumns": 0, "variableReplacements": [], "showCorrectAnswer": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "choices": ["<p>AND</p>", "<p>OR</p>"], "variableReplacementStrategy": "originalfirst", "distractors": ["", ""], "maxMarks": 0, "marks": 0, "scripts": {}, "displayType": "dropdownlist", "type": "1_n_2", "minMarks": 0, "matrix": ["1", 0], "extendBaseMarkingAlgorithm": true, "shuffleChoices": false}], "marks": 0, "prompt": "<p>Find the range of vaues such that:&nbsp;$\\simplify{({g}+{f})x-{g}{f}&gt;x^2}$ &nbsp;&nbsp;</p>\n<p>Rearrange&nbsp;then factorise&nbsp;the inequality:</p>\n<p>[[0]]&nbsp;$&lt;0$.</p>\n<p>Use the above result&nbsp;to find the range&nbsp;of values for $x$ such that $\\simplify{({g}+{f})x-{g}{f}&gt;x^2}$.</p>\n<p>$x&gt;$&nbsp;[[1]]&nbsp;</p>\n<p>[[3]]&nbsp;&nbsp;</p>\n<p>$x&lt;$&nbsp;[[2]]</p>", "scripts": {}, "type": "gapfill", "extendBaseMarkingAlgorithm": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}]}