// Numbas version: exam_results_page_options
{"name": "Complete the square. (Video)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Complete the square. (Video)", "tags": ["Steps", "algebra", "algebraic manipulation", "complete the square", "completing the square", "quadratics", "roots of a polynomial", "roots of a quadratic", "solving equations", "solving quadratic equation", "steps", "video"], "advice": "\n  \n  \n  <p>Completing the square for the quadratic expression $\\simplify{{a*b}x^2+{-n1}x+{c*d}}$<br/>\\[\\begin{eqnarray}\n  \n  \\simplify{{a*b}x^2+{-n1}x+{c*d}}&#38;=&#38;\\var{n5}\\left(\\simplify{x^2+({-n1}/{a*b})x+ {c*d}/{a*b}}\\right)\\\\\n  \n  &#38;=&#38;\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2+ \\simplify{{c*d}/{a*b}-({-n1}/({2*a*b}))^2}\\right)\\\\\n  \n  &#38;=&#38;\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2 -\\simplify{ {n2^2}/{4*(a*b)^2}}\\right)\\\\\n  \n  &#38;=&#38;\\var{n5}\\left(\\simplify{x+({-n1}/{2*n5})}\\right)^2 -\\simplify{ {n2^2}/{4*(n5)}}\n  \n  \\end{eqnarray}\n  \n  \\]<br/>So to solve $\\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$ we have to solve:<br/>\\[\\begin{eqnarray}\n  \n  \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&#38;\\phantom{{}}&#38; -\\simplify{ {n2^2}/{4*(a*b)^2}}=0\\Rightarrow\\\\\n  \n  \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&#38;\\phantom{{}}&#38;=\\simplify{ {n2^2}/{4*(a*b)^2}=({abs(n2)}/{2*a*b})^2}\n  \n  \\end{eqnarray}\\]<br/>So we get the two {rep} solutions:<br/>\\[\\begin{eqnarray}\n  \n  \\simplify{x+({-n1}/{2*a*b})}&#38;=&#38;\\simplify{{abs(n2)}/{2*a*b}} \\Rightarrow  &#38;x&#38; = \\simplify{({abs(n2)+n1}/{2*a*b})}\\\\\n  \n  \\simplify{x+({-n1}/{2*a*b})}&#38;=&#38;\\simplify{-({abs(n2)}/{2*a*b})} \\Rightarrow &#38;x&#38; = \\simplify{({n1-abs(n2)}/{2*a*b})}\n  \n  \\end{eqnarray}\\]</p>\n  \n  \n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "prompt": "<p>\\[q(x)=\\simplify[std]{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}\\]<br/>Write $q(x) = a(x+b)^2+c\\;\\;$ for fractions or integers $a$, $b$, $c$.</p>\n        <p>$q(x)=\\;$ [[0]]</p>\n        <p>You can get more information on completing the square by clicking on Show steps. You will lose 1 mark if you do so.</p>\n        <p>You will also find a video covering the use of completing the square in solving a quadratic equation.</p>\n        <p>Input all numbers as fractions or integers and not as decimals.</p>", "gaps": [{"notallowed": {"message": "<p>Write in the form $a(x+b)^2+c$ without using decimals.</p>", "showstrings": false, "strings": [".", "x*x", "x x", "x(", "x (", ")x", ") x"], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 2.0, "answer": "{n5}(x+({-n1}/{2*n5}))^2-{n2^2}/{4*n5}", "type": "jme", "musthave": {"message": "<p>write in the form $a(x+b)^2+c$</p>", "showstrings": false, "strings": ["(", ")", "^"], "partialcredit": 0.0}}], "steps": [{"prompt": "<p>Given the quadratic $\\simplify{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}$ we complete the square by:</p>\n          <p>1. Writing the quadratic as \\[\\var{n5}\\left(\\simplify{x^2+({-n1}/{n5})x+ {c*d}/{n5}}\\right)\\]</p>\n          <p>2. Then complete the square for the quadratic \\[\\simplify{x^2+({-n1}/{n5})x+ {c*d}/{n5}}\\]</p>\n          <p>3. Remember to multiply by $\\var{n5}$ the expression found from the second stage.</p>\n          <p>The following video steps through a similar problem.</p>\n          <p><iframe src=\"https://player.vimeo.com/video/8173863\" frameborder=\"0\" width=\"640\" height=\"480\"></iframe></p>\n          <p>&nbsp;</p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}, {"prompt": "\n      <p>Now find the roots of the quadratic equation $\\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$.</p>\n        <p>The least root is $x=\\;$ [[0]]. The greatest root is $x=\\;$ [[1]].</p>\n        <p>Input the roots as fractions or integers not as decimals.</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input numbers as fractions or integers not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{n1-n4}/{2*a*b}", "type": "jme"}, {"notallowed": {"message": "<p>Input numbers as fractions or integers not as &nbsp;decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{n1+n4}/{2*a*b}", "type": "jme"}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "<p>Complete the square for the quadratic expression $q(x)$ given below by writing it in the form \\[a(x+b)^2+c\\] for numbers $a,\\;b$ and $c$.</p>\n  <p>Hence, or otherwise, find both roots of the equation $q(x)=0$.</p>\n  <p>&nbsp;</p>", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..5)", "name": "a"}, "c": {"definition": "c1*s3", "name": "c"}, "b": {"definition": "random(1..4)", "name": "b"}, "d": {"definition": "if(d1=b*c/a, max(d1+1,random(1..5))*s3,d1*s3)", "name": "d"}, "f": {"definition": "a*b", "name": "f"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "n5": {"definition": "a*b", "name": "n5"}, "disc": {"definition": "(b*c+a*d)^2-4*a*b*c*d", "name": "disc"}, "c1": {"definition": "switch(f=1, random(1..6),f=2,random(1,3,5,7,9),f=3,random(1,2,5,7,8),f=4,random(1,3,5,7,9),f=6, random(1,5,7,8),f=9,random(1,2,4,7,8),f=8,random(1,3,5,7,9),f=12,random(1,5,7),random(1,3,5,7))", "name": "c1"}, "rep": {"definition": "switch(disc=0,'repeated', ' ')", "name": "rep"}, "n1": {"definition": "b*c+a*d", "name": "n1"}, "n2": {"definition": "b*c-a*d", "name": "n2"}, "n3": {"definition": "2*a*b", "name": "n3"}, "n4": {"definition": "abs(n2)", "name": "n4"}, "rdis": {"definition": "switch(disc=0,'The discriminant is '+ 0+' and so we get two repeated roots in this case.',disc<0, 'There are no real roots.','The roots exist and are distinct. ')", "name": "rdis"}, "d1": {"definition": "switch(f=1, random(1..6),f=2,random(1,3,5,7,9),f=3,random(1,2,5,7,8),f=4,random(1,3,5,7,9),f=6, random(1,5,7,8),f=9,random(1,2,4,7,8),f=8,random(1,3,5,7,9),f=12,random(1,5,7),random(1,3,5,7))", "name": "d1"}}, "metadata": {"notes": "\n                            \t\t<p><strong>5/08/2012:</strong></p>\n                            \t\t    <p>Added tags.</p>\n                            \t\t    <p>Added description.</p>\n                            \t\t    <p>Corrected variable value n2 to ensure that there are no repeated roots.</p>\n                            \t\t    <p>Checked calculation.OK.</p>\n                            \t\t    <p>Improved display in content areas.</p>\n                            \t\t", "description": "<p>Find $p$ and $q$ such that $ax^2+bx+c = a(x+p)^2+q$.&nbsp;</p>\n                        <p>Hence, or otherwise, find roots of &nbsp;$ax^2+bx+c=0$.</p>\n                        <p>Includes a video which shows how to solve a quadratic by completing the square.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}