// Numbas version: exam_results_page_options
{"name": "Use the quadratic formula to solve an equation in terms of an unknown variable", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [{"variables": ["b", "c", "n2", "b_2", "c_2", "b_3"], "name": "part 2"}], "tags": [], "parts": [{"variableReplacementStrategy": "originalfirst", "steps": [{"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "unitTests": [], "type": "information", "prompt": "<p>The quadratic formula is</p>\n<p>\\[{\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\\]</p>"}], "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "stepsPenalty": 0, "gaps": [{"variableReplacementStrategy": "originalfirst", "customName": "", "failureRate": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingType": "absdiff", "variableReplacements": [], "valuegenerators": [{"name": "k", "value": ""}], "scripts": {}, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "vsetRange": [0, 1], "useCustomName": false, "answer": "({-b_3/2-sqrt(b_3^2-4*c_2)/2})k", "unitTests": [], "type": "jme", "showPreview": true, "checkVariableNames": false}, {"variableReplacementStrategy": "originalfirst", "customName": "", "failureRate": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingType": "absdiff", "variableReplacements": [], "valuegenerators": [{"name": "k", "value": ""}], "scripts": {}, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "vsetRange": [0, 1], "useCustomName": false, "answer": "({-b_3/2+sqrt(b_3^2-4*c_2)/2})k", "unitTests": [], "type": "jme", "showPreview": true, "checkVariableNames": false}], "unitTests": [], "type": "gapfill", "prompt": "<p>Solve the equation $\\simplify {x^2+{b_3}*k*x+{c_2}k^2=0}$. Give your answer in terms of $k$. Assuming $k$ is positive, enter the lowest root first.</p>\n<p>$x_1=$&nbsp;[[0]]</p>\n<p>$x_2=$&nbsp;[[1]]</p>"}], "functions": {}, "name": "Use the quadratic formula to solve an equation in terms of an unknown variable", "ungrouped_variables": ["a1", "a2", "a3", "a4", "b1", "b2", "b3", "b4", "x1", "p1", "p2", "x2", "a", "m"], "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Factorise a quadratic expression of the form $x^2+akx+bk^2$ for $x$, in terms of $k$. $a$ and $b$ are constants.</p>"}, "extensions": [], "statement": "<p>The coefficients of the following equation involve the unknown value $k$. We can use the quadratic formula to find expressions for the values of $x$ in terms of $k$.</p>", "advice": "<p>The quadratic formula is</p>\n<p>\\[{\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\\]</p>\n<p>We can list our values for $a, b$ and $c$.</p>\n<p>\\[\\begin{align}<br/>a&amp;=1\\\\<br/>b&amp;=\\var{b_3}k\\\\<br/>c&amp;=\\var{c_2}k^2<br/>\\end{align}\\]</p>\n<p>Then by substituting them into the quadratic formula, we obtain</p>\n<p>\\[x=\\frac {-\\var{b_3}k\\pm\\sqrt{\\var{b_3}^2k^2-4\\times \\var{c_2}k^2}}{2}\\]</p>\n<p>We can then simplify this equation to</p>\n<p>\\[\\begin{align}<br/>x&amp;=\\frac {-\\var{b_3}k\\pm k\\sqrt{\\var{b_3}^2-\\var{4c_2}}}{2}\\\\<br/>\\end{align}\\]<br/>\\[\\begin{align}<br/>&amp;=k\\left(\\frac{-\\var{b_3}}{2}\\pm&nbsp;\\frac {\\sqrt{\\var{b_3}^2-\\var{4c_2}}}{2}\\right)\\\\<br/>\\end{align}\\]<br/>\\[\\begin{align}<br/>&amp;=k\\left(-\\frac{\\var{b_3}}{2} \\pm\\frac{\\sqrt{\\var{(b_3^2-4c_2)}}}{2}\\right)\\\\<br/>\\end{align}\\]</p>\n<p>This means our possible values for $x$ in terms of $k$ are,</p>\n<p>\\begin{align}<br/>x_1 &amp;= \\left( \\simplify[all,!noleadingminus,!collectnumbers,!simplifyfractions]{-{b_3}/2 - {sqrt(b_3^2-4c_2)}/2} \\right) k = \\var[fractionnumbers]{-b_3/2 - sqrt(b_3^2-4*c_2)/2}k \\\\<br/>x_2 &amp;= \\left( \\simplify[all,!noleadingminus,!collectnumbers,!simplifyfractions]{-{b_3}/2 +&nbsp;{sqrt(b_3^2-4c_2)}/2} \\right) k = \\var[fractionnumbers]{-b_3/2 + sqrt(b_3^2-4*c_2)/2}k<br/>\\end{align}</p>", "variables": {"b_2": {"description": "", "definition": "random(n2^0.5..11 except n2^0.5)^2 ", "group": "part 2", "name": "b_2", "templateType": "anything"}, "b3": {"description": "", "definition": "random(-30..6 except a3)", "group": "Ungrouped variables", "name": "b3", "templateType": "anything"}, "x2": {"description": "", "definition": "(-(a2)+((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)", "group": "Ungrouped variables", "name": "x2", "templateType": "anything"}, "c": {"description": "", "definition": "random(-10..4 except 0)^2", "group": "part 2", "name": "c", "templateType": "anything"}, "b": {"description": "", "definition": "2c^0.5+a1^2", "group": "part 2", "name": "b", "templateType": "anything"}, "b2": {"description": "", "definition": "random(20..35 except a2)", "group": "Ungrouped variables", "name": "b2", "templateType": "anything"}, "a": {"description": "", "definition": "random(15..20)", "group": "Ungrouped variables", "name": "a", "templateType": "anything"}, "a4": {"description": "", "definition": "random(1..15)", "group": "Ungrouped variables", "name": "a4", "templateType": "anything"}, "b_3": {"description": "", "definition": "b_2^0.5", "group": "part 2", "name": "b_3", "templateType": "anything"}, "a2": {"description": "", "definition": "random(9..30 except a1)", "group": "Ungrouped variables", "name": "a2", "templateType": "anything"}, "c_2": {"description": "", "definition": "(b_2-n2)/4", "group": "part 2", "name": "c_2", "templateType": "anything"}, "p2": {"description": "", "definition": "(-(b2-b4)+((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "group": "Ungrouped variables", "name": "p2", "templateType": "anything"}, "b4": {"description": "", "definition": "random(-5..5)", "group": "Ungrouped variables", "name": "b4", "templateType": "anything"}, "p1": {"description": "", "definition": "(-(b2-b4)-((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "group": "Ungrouped variables", "name": "p1", "templateType": "anything"}, "b1": {"description": "", "definition": "random(1..10 except a1)", "group": "Ungrouped variables", "name": "b1", "templateType": "anything"}, "m": {"description": "", "definition": "random(9..a)", "group": "Ungrouped variables", "name": "m", "templateType": "anything"}, "a1": {"description": "", "definition": "random(1..9)", "group": "Ungrouped variables", "name": "a1", "templateType": "anything"}, "x1": {"description": "", "definition": "(-(a2)-((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)", "group": "Ungrouped variables", "name": "x1", "templateType": "anything"}, "n2": {"description": "", "definition": "random(1..10)^2", "group": "part 2", "name": "n2", "templateType": "anything"}, "a3": {"description": "", "definition": "random(-30..4 except 0)", "group": "Ungrouped variables", "name": "a3", "templateType": "anything"}}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}