// Numbas version: exam_results_page_options
{"name": "Solve a quadratic with the a-b-c-formula (the quadratic formula) ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "f", "s3", "s2", "s1", "n4", "n2", "disc", "rdis", "n1", "rep", "n3", "c1", "n5", "d1"], "name": "Solve a quadratic with the a-b-c-formula (the quadratic formula) ", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p><em></em></p>\n<p><strong></strong></p>\n<p><strong>Direct Factorisation</strong></p>\n<p>If you can spot a direct factorisation then this is the quickest way to do this question.</p>\n<p>For this example we have the factorisation</p>\n<p>\\[\\simplify{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d} = ({a} * x + { -c}) * ({b} * x + { -d})}\\]</p>\n<p>Hence we find the roots:<br/>\\[\\begin{eqnarray} x&amp;=&amp; \\simplify{{n1-n4}/{2*a*b}}\\\\ x&amp;=&amp; \\simplify{{n1+n4}/{2*a*b}} \\end{eqnarray} \\]</p>\n<p><strong>Other Methods.</strong></p>\n<p>There are several methods of finding the roots &ndash; here are the main methods.</p>\n<p><strong>Finding the roots of a quadratic using the standard formula.</strong></p>\n<p>We can use the following formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$</p>\n<p>The two roots are</p>\n<p>\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]<br/>there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$</p>\n<p>1. $\\Delta \\gt 0$. The roots are real and distinct</p>\n<p>2. $\\Delta=0$. The roots are real and equal. Their common value is $\\displaystyle -\\frac{b}{2a}$</p>\n<p>3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.</p>\n<p>For this question the discriminant of $\\simplify{{a*b}x^2+{-b*c-a*d}x+{c*d}}$ is $\\Delta = \\simplify[std]{{-n1}^2-4*{a*b*c*d}}=\\var{disc}$</p>\n<p>{rdis}.</p>\n<p>So the {rep} roots are:</p>\n<p>\\[\\begin{eqnarray} x = \\frac{\\var{n1} - \\sqrt{\\var{disc}}}{\\var{n3}} &amp;=&amp; \\frac{\\var{n1} - \\var{n4} }{\\var{n3}} &amp;=&amp; \\simplify{{n1 - n4}/ {n3}}\\\\ x = \\frac{\\var{n1} + \\sqrt{\\var{disc}}}{\\var{n3}} &amp;=&amp; \\frac{\\var{n1} + \\var{n4} }{\\var{n3}} &amp;=&amp; \\simplify{{n1 + n4}/ {n3}} \\end{eqnarray}\\]</p>\n<p><strong>Completing the square.</strong></p>\n<p>First we complete the square for the quadratic expression $\\simplify{{a*b}x^2+{-n1}x+{c*d}}$<br/>\\[\\begin{eqnarray} \\simplify{{a*b}x^2+{-n1}x+{c*d}}&amp;=&amp;\\var{n5}\\left(\\simplify{x^2+({-n1}/{a*b})x+ {c*d}/{a*b}}\\right)\\\\ &amp;=&amp;\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2+ \\simplify{{c*d}/{a*b}-({-n1}/({2*a*b}))^2}\\right)\\\\ &amp;=&amp;\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2 -\\simplify{ {n2^2}/{4*(a*b)^2}}\\right) \\end{eqnarray} \\]<br/>So to solve $\\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$ we have to solve:<br/>\\[\\begin{eqnarray} \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&amp;\\phantom{{}}&amp; -\\simplify{ {n2^2}/{4*(a*b)^2}}=0\\Rightarrow\\\\ \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&amp;\\phantom{{}}&amp;=\\simplify{ {n2^2}/{4*(a*b)^2}=({abs(n2)}/{2*a*b})^2} \\end{eqnarray}\\]<br/>So we get the two {rep} solutions:<br/>\\[\\begin{eqnarray} \\simplify{x+({-n1}/{2*a*b})}&amp;=&amp;\\simplify{-{abs(n2)}/{2*a*b}} \\Rightarrow &amp;x&amp; = \\simplify{({-abs(n2)+n1}/{2*a*b})}\\\\ \\simplify{x+({-n1}/{2*a*b})}&amp;=&amp;\\simplify{({abs(n2)}/{2*a*b})} \\Rightarrow &amp;x&amp; = \\simplify{({n1+abs(n2)}/{2*a*b})} \\end{eqnarray}\\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "<p>Solve for $x$: &nbsp;\\[\\simplify[std]{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d}}=0\\]<br/>The least root is $x=\\;$ [[0]]. The greatest root is $x=\\;$ [[1]].</p>\n<p>You can get more information on solving a quadratic by clicking on \"Show steps\". You will lose 1 mark if you do so.</p>\n<p>Enter the least root first. If the roots are equal, enter the root in both input boxes.</p>\n<p>Enter the roots as fractions or integers, not as decimals.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "\n\t\t\t\t\t          <p><strong>Finding the roots by factorisation.</strong></p>\n\t\t\t\t\t          <p>Finding a factorisation of a quadratic $q(x)=a(x-r)(x-s)$ where $a$ is the coefficient of $x^2$ gives the roots $x=r$, $x=s$ immendiately.</p>\n\t\t\t\t\t          <p>If you cannot find a factorisation then there are several other methods you can use.</p>\n\t\t\t\t\t          <p><strong>Using the formula for the roots.</strong></p>\n\t\t\t\t\t          <p>You can find the roots by using the formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$</p>\n\t\t\t\t\t          <p>The two roots are:</p>\n\t\t\t\t\t          <p>\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]<br/>there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$</p>\n\t\t\t\t\t          <p>1. $\\Delta \\gt 0$. The roots are real and distinct</p>\n\t\t\t\t\t          <p>2. $\\Delta=0$. The roots are real and equal. Their value is $\\displaystyle \\frac{-b}{2a}$</p>\n\t\t\t\t\t          <p>3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.</p>\n\t\t\t\t\t          <p>&nbsp;</p>\n\t\t\t\t\t          \n\t\t\t\t\t            \n\t\t\t\t\t            \n\t\t\t\t\t          \n\t\t\t\t\t          \n\t\t\t\t\t          \n\t\t\t\t\t          \n\t\t\t\t\t          \n\t\t\t\t\t          \n\t\t\t\t\t            \n\t\t\t\t\t          \n\t\t\t\t\t", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"notallowed": {"message": "<p>Input numbers as fractions or integers not as a decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.0001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{n1-n4}/{2*a*b}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"notallowed": {"message": "<p>Input numbers as fractions or integers not as a decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.0001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{n1+n4}/{2*a*b}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>Find the roots of the following quadratic equation.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "c1*s3", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "if((a*d1)^2=(b*c)^2, max(d1+1,random(1..5))*s3,d1*s3)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "f": {"definition": "a*b", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "n5": {"definition": "a*b", "templateType": "anything", "group": "Ungrouped variables", "name": "n5", "description": ""}, "disc": {"definition": "(b*c+a*d)^2-4*a*b*c*d", "templateType": "anything", "group": "Ungrouped variables", "name": "disc", "description": ""}, "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))", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "rep": {"definition": "switch(disc=0,'repeated', ' ')", "templateType": "anything", "group": "Ungrouped variables", "name": "rep", "description": ""}, "n1": {"definition": "b*c+a*d", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "n2": {"definition": "b*c-a*d", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}, "n3": {"definition": "2*a*b", "templateType": "anything", "group": "Ungrouped variables", "name": "n3", "description": ""}, "n4": {"definition": "abs(n2)", "templateType": "anything", "group": "Ungrouped variables", "name": "n4", "description": ""}, "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. ')", "templateType": "anything", "group": "Ungrouped variables", "name": "rdis", "description": ""}, "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))", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}}, "metadata": {"description": "<p>Solve for $x$: $\\displaystyle ax ^ 2 + bx + c=0$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Richard Miles", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/882/"}]}]}], "contributors": [{"name": "Richard Miles", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/882/"}]}