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