// Numbas version: exam_results_page_options
{"name": "Simon's copy of 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": [{"variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"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>", "licence": "Creative Commons Attribution 4.0 International"}, "tags": ["Quadratic equations", "Quadratic Equations", "quadratic equations", "quadratic formula", "Quadratic formula", "solving quadratic equations", "Solving quadratic equations", "Solving quadratic equations using the quadratic formula", "taxonomy", "unknown variable", "Unknown variable"], "extensions": [], "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": {"b1": {"definition": "random(1..10 except a1)", "name": "b1", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a1": {"definition": "random(1..9)", "name": "a1", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a": {"definition": "random(15..20)", "name": "a", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "n2": {"definition": "random(1..10)^2", "name": "n2", "templateType": "anything", "group": "part 2", "description": ""}, "b2": {"definition": "random(20..35 except a2)", "name": "b2", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "c": {"definition": "random(-10..4 except 0)^2", "name": "c", "templateType": "anything", "group": "part 2", "description": ""}, "p2": {"definition": "(-(b2-b4)+((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "name": "p2", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "m": {"definition": "random(9..a)", "name": "m", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b_3": {"definition": "b_2^0.5", "name": "b_3", "templateType": "anything", "group": "part 2", "description": ""}, "a2": {"definition": "random(9..30 except a1)", "name": "a2", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "c_2": {"definition": "(b_2-n2)/4", "name": "c_2", "templateType": "anything", "group": "part 2", "description": ""}, "a4": {"definition": "random(1..15)", "name": "a4", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "a3": {"definition": "random(-30..4 except 0)", "name": "a3", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b": {"definition": "2c^0.5+a1^2", "name": "b", "templateType": "anything", "group": "part 2", "description": ""}, "b4": {"definition": "random(-5..5)", "name": "b4", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "p1": {"definition": "(-(b2-b4)-((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "name": "p1", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "x2": {"definition": "(-(a2)+((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)", "name": "x2", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "x1": {"definition": "(-(a2)-((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)", "name": "x1", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "b_2": {"definition": "random(n2^0.5..11 except n2^0.5)^2 ", "name": "b_2", "templateType": "anything", "group": "part 2", "description": ""}, "b3": {"definition": "random(-30..6 except a3)", "name": "b3", "templateType": "anything", "group": "Ungrouped variables", "description": ""}}, "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a1", "a2", "a3", "a4", "b1", "b2", "b3", "b4", "x1", "p1", "p2", "x2", "a", "m"], "parts": [{"variableReplacements": [], "steps": [{"variableReplacements": [], "marks": 0, "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "prompt": "<p>The quadratic formula is</p>\n<p>\\[{\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\\]</p>"}], "marks": 0, "stepsPenalty": 0, "gaps": [{"variableReplacements": [], "checkingaccuracy": 0.001, "checkvariablenames": false, "vsetrange": [0, 1], "type": "jme", "vsetrangepoints": 5, "scripts": {}, "marks": 1, "showpreview": true, "answer": "({-b_3/2-sqrt(b_3^2-4*c_2)/2})k", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showCorrectAnswer": true, "checkingtype": "absdiff"}, {"variableReplacements": [], "checkingaccuracy": 0.001, "checkvariablenames": false, "vsetrange": [0, 1], "type": "jme", "vsetrangepoints": 5, "scripts": {}, "marks": 1, "showpreview": true, "answer": "({-b_3/2+sqrt(b_3^2-4*c_2)/2})k", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "showCorrectAnswer": true, "checkingtype": "absdiff"}], "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "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>"}], "rulesets": {}, "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>", "name": "Simon's copy of Use the quadratic formula to solve an equation in terms of an unknown variable", "variable_groups": [{"name": "part 2", "variables": ["b", "c", "n2", "b_2", "c_2", "b_3"]}], "functions": {}, "type": "question", "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": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "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": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}