// Numbas version: exam_results_page_options {"name": "Year 11 Algebra quiz - Testing", "metadata": {"description": "

This quiz contains questions on algebraic fractions, logarithmic equations, exponential equations, quadratic equations and simultaneous equations.

", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 1800, "percentPass": "80", "showQuestionGroupNames": true, "showstudentname": true, "question_groups": [{"name": "Year 11MAB", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", "", "", ""], "questions": [{"name": "Evaluate f(x)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "parts": [{"type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"maxValue": "{a1}*({a3}^{a2})+{b1}*{a3}-{c1}", "variableReplacements": [], "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "strictPrecision": false, "precisionType": "dp", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "type": "numberentry", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "2", "minValue": "{a1}*({a3}^{a2})+{b1}*{a3}-{c1}", "showPrecisionHint": true, "allowFractions": false, "showCorrectAnswer": true, "scripts": {}, "mustBeReduced": false, "marks": 1}], "showFeedbackIcon": true, "prompt": "

Evaluate \$$f(\\var{a3})\$$

\n

\$$f(\\var{a3})\$$ = [[0]]

", "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0}], "rulesets": {}, "functions": {}, "ungrouped_variables": ["a1", "a2", "b1", "c1", "a3"], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Evaluating a function

"}, "preamble": {"css": "", "js": ""}, "variable_groups": [], "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a1": {"definition": "random(2..12#1)", "description": "", "group": "Ungrouped variables", "name": "a1", "templateType": "randrange"}, "a3": {"definition": "random(1..6#1)", "description": "", "group": "Ungrouped variables", "name": "a3", "templateType": "randrange"}, "c1": {"definition": "random(1..15#1)", "description": "", "group": "Ungrouped variables", "name": "c1", "templateType": "randrange"}, "a2": {"definition": "random(2..5#1)", "description": "", "group": "Ungrouped variables", "name": "a2", "templateType": "randrange"}, "b1": {"definition": "random(3..12#1)", "description": "", "group": "Ungrouped variables", "name": "b1", "templateType": "randrange"}}, "advice": "

\$$f(x)=\\var{a1}x^{\\var{a2}}+\\var{b1}x-\\var{c1}\$$

\n

\$$x=\\var{a3}\$$

\n

\$$f(\\var{a3})=\\var{a1}*(\\var{a3})^{\\var{a2}}+\\var{b1}*(\\var{a3})-\\var{c1}\$$

\n

\$$f(\\var{a3})=\\simplify{{a1}*{a3}^{{a2}}}+\\simplify{{b1}*{a3}}-\\var{c1}\$$

\n

\$$f(\\var{a3})=\\simplify{{a1}*{a3}^{{a2}}+{b1}*{a3}-{c1}}\$$

", "statement": "

Given the function:

\n

\$$f(x)=\\var{a1}x^{\\var{a2}}+\\var{b1}x-\\var{c1}\$$

", "type": "question"}, {"name": "Jill's copy of Jill's copy of Manipulation of formula 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Jill Singleton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2473/"}], "metadata": {"description": "

Manipulation of algebraic fractions

", "licence": "Creative Commons Attribution 4.0 International"}, "tags": [], "ungrouped_variables": ["a", "b", "c", "d"], "variables": {"c": {"description": "", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "c"}, "b": {"description": "", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "b"}, "a": {"description": "", "definition": "random(6..15#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a"}, "d": {"description": "", "definition": "random(8..16#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "d"}}, "variable_groups": [], "variablesTest": {"condition": "{a}*{d}>{c}*{b}", "maxRuns": 100}, "rulesets": {}, "parts": [{"type": "gapfill", "prompt": "

\n

\$$V =\$$ [[0]]

", "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"type": "jme", "answer": "((5*{b}-{d})R+8)/(({a}*{d}-{b}*{c})R+7*{a}-3*{c})", "scripts": {}, "showpreview": true, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showFeedbackIcon": true, "checkvariablenames": false, "showCorrectAnswer": true, "variableReplacements": [], "checkingaccuracy": 0.001, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": "2"}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": []}], "preamble": {"css": "", "js": ""}, "functions": {}, "advice": "

When one fraction equals another fraction we can clear both fractions by cross-multiplying:

\n

\$$(\\var{a}V+1)*(\\var{d}R+7)=(\\var{b}R+3)*(\\var{c}V+5)\$$

\n

\$$\\simplify{{a}*{d}}VR+\\simplify{7*{a}}V+\\var{d}R+7=\\simplify{{b}*{c}}VR+\\simplify{5*{b}}R+\\simplify{3*{c}}V+15\$$

\n

Gathering all the terms involving \$$V\$$ to the left hand side and moving all other terms to the right hand side gives

\n

\$$\\simplify{{a}*{d}-{b}*{c}}VR+\\simplify{7*{a}-3*{c}}V=\\simplify{5*{b}-{d}}R+8\$$

\n

Factoring \$$V\$$ out on the left hand side

\n

\$$V(\\simplify{{a}*{d}-{b}*{c}}R+\\simplify{7*{a}-3*{c}})=\\simplify{5*{b}-{d}}R+8\$$

\n

Thus

\n

\$$V=\\frac{\\simplify{5*{b}-{d}}R+8}{\\simplify{{a}*{d}-{b}*{c}}R+\\simplify{7*{a}-3*{c}}}\$$

", "statement": "

Rearrange the following expression to make V the subject:

\n

\$$\\frac{\\var{a}V+1}{\\var{b}R+3}=\\frac{\\var{c}V+5}{\\var{d}R+7}\$$

", "type": "question"}, {"name": "Manipulation of formula 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "metadata": {"description": "

Manipulation of an exponential function

", "licence": "Creative Commons Attribution 4.0 International"}, "tags": [], "ungrouped_variables": ["k", "c", "m", "d"], "variables": {"c": {"description": "", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "c"}, "m": {"description": "", "definition": "random(1.6..5#0.2)", "templateType": "randrange", "group": "Ungrouped variables", "name": "m"}, "k": {"description": "", "definition": "random(2..12#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "k"}, "d": {"description": "", "definition": "random(1..14#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "d"}}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "parts": [{"type": "gapfill", "prompt": "

\$$x =\$$ [[0]]

", "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"type": "jme", "answer": "(ln((1-y/{k})/{c})-{d})/{m}", "scripts": {}, "showpreview": true, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showFeedbackIcon": true, "checkvariablenames": false, "showCorrectAnswer": true, "variableReplacements": [], "checkingaccuracy": 0.001, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": "2"}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": []}], "preamble": {"css": "", "js": ""}, "functions": {}, "advice": "

\$$y=\\var{k}(1-\\var{c}e^{\\var{m}x+\\var{d}})\$$

\n

Working from the outside in, we divide across by \$$\\var{k}\$$

\n

\$$\\frac{y}{\\var{k}}=1-\\var{c}e^{\\var{m}x+\\var{d}}\$$

\n

We can bring the \$$x\$$ variable to the left hand side and move the \$$y\$$ variable to the right hand side

\n

\$$\\var{c}e^{\\var{m}x+\\var{d}}=1-\\frac{y}{\\var{k}}\$$

\n

Again working from the outside in we divide across by \$$\\var{c}\$$

\n

\$$e^{\\var{m}x+\\var{d}}=\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\$$

\n

Taking the natural log of both sides eliminates the \$$e\$$ from the left hand side.

\n

\$$\\var{m}x+\\var{d}=ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)\$$

\n

Subtract \$$\\var{d}\$$ from both sides

\n

\$$\\var{m}x=ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)-\\var{d}\$$

\n

and finally divide by \$$\\var{m}\$$ to get

\n

\$$x=\\frac{ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)-\\var{d}}{\\var{m}}\$$

", "statement": "

Rearrange the following expression to make \$$x\$$ the subject:

\n

\$$y=\\var{k}(1-\\var{c}e^{\\var{m}x+\\var{d}})\$$

", "type": "question"}, {"name": "Solve a logarithmic equation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "parts": [{"scripts": {}, "marks": 0, "variableReplacements": [], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

Calculate the value of \$$x\$$ that satisfies the equation when  \$$y=\\var{d}\$$.

\n

\n

\$$x = \$$ [[0]]

", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "((10^(d/a))-c)/b", "mustBeReduced": false, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "allowFractions": false, "variableReplacements": [], "strictPrecision": false, "marks": "2", "minValue": "((10^(d/a))-c)/b", "precisionType": "dp", "showPrecisionHint": false, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "precision": "3"}]}], "statement": "

Given the following logarithmic equation:

\n

\$$y=\\var{a}log(\\var{b}x+\\var{c}))\$$

\n

", "rulesets": {}, "variable_groups": [], "ungrouped_variables": ["a", "b", "c", "d"], "functions": {}, "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Solve a logarithmic equation

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

\$$\\var{a}log(\\var{b}x+\\var{c})=\\var{d}\$$

\n

Divide across by \$$\\var{a}\$$

\n

\$$log(\\var{b}x+\\var{c})=\\var{d}/\\var{a}=\\simplify{{d}/{a}}\$$

\n

\$$\\var{b}x+\\var{c}=10^{\\simplify{{d}/{a}}}\$$

\n

\$$\\var{b}x+\\var{c}=\\simplify{10^{{d}/{a}}}\$$

\n

\$$\\var{b}x=\\simplify{10^{{d}/{a}}}-\\var{c}\$$

\n

\$$\\var{b}x=\\simplify{10^{{d}/{a}}-{c}}\$$

\n

\$$x=\\simplify{(10^{{d}/{a}}-{c})/{b}}\$$

", "preamble": {"css": "", "js": ""}, "variables": {"b": {"name": "b", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..6#1)"}, "c": {"name": "c", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..10#1)"}, "a": {"name": "a", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..8#1)"}, "d": {"name": "d", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(1..6#1)"}}, "type": "question"}, {"name": "Solve an exponential equation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "advice": "

\$$\\var{k}=\\var{a}e^{\\var{m}x+{\\var{c}}}\$$

\n

\$$\\frac{\\var{k}}{\\var{a}}=e^{\\var{m}x+\\var{c}}\$$

\n

\$$ln\\left(\\frac{\\var{k}}{\\var{a}}\\right)=\\var{m}x+\\var{c}\$$

\n

\$$ln\\left(\\frac{\\var{k}}{\\var{a}}\\right)-\\var{c}=\\var{m}x\$$

\n

\$$\\frac{ln\\left(\\frac{\\var{k}}{\\var{a}}\\right)-\\var{c}}{\\var{m}}=x\$$

Solve an exponential equation

"}, "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a", "m", "c", "k"], "variable_groups": [], "functions": {}, "statement": "

Given the equation \$$f(x)=\\var{a}e^{\\var{m}x+\\var{c}}\$$

\n

Determine the value for \$$x\$$ that satisfies the relation \$$f(x)=\\var{k}\$$

", "tags": [], "rulesets": {}, "parts": [{"marks": 0, "prompt": "

\n

\$$x = \$$ [[0]]

", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "type": "gapfill", "gaps": [{"mustBeReducedPC": 0, "precisionPartialCredit": 0, "maxValue": "(ln({k}/{a})-{c})/{m}", "precision": "3", "strictPrecision": false, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "type": "numberentry", "allowFractions": false, "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "showPrecisionHint": false, "marks": "2", "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "minValue": "(ln({k}/{a})-{c})/{m}"}]}], "variables": {"a": {"templateType": "randrange", "group": "Ungrouped variables", "name": "a", "description": "", "definition": "random(4..20#1)"}, "c": {"templateType": "randrange", "group": "Ungrouped variables", "name": "c", "description": "", "definition": "random(0.1..3#0.2)"}, "m": {"templateType": "randrange", "group": "Ungrouped variables", "name": "m", "description": "", "definition": "random(0.1..1.5#0.1)"}, "k": {"templateType": "randrange", "group": "Ungrouped variables", "name": "k", "description": "", "definition": "random(100..150#1)"}}, "variablesTest": {"maxRuns": 100, "condition": "c>0 or c<0 "}, "type": "question"}, {"name": "Solving quadratic equations 1(a)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "statement": "

There are two values that satisfy the quadratic equation:

\n

\$$\\var{a1}x^2+\\simplify{{{a1}*{b1}*{c1}}}=\\simplify{{a1}*{b1}+{a1}{c1}}x\$$

", "rulesets": {}, "variable_groups": [], "tags": [], "functions": {}, "parts": [{"scripts": {}, "marks": 0, "variableReplacements": [], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

Type in the greater of the two values that satisfies the equation.

\n

Input your answer correct to three decimal places.  \$$x = \$$ [[0]]

\n

Type in the lesser of the two values that satisfies the equation.

\n

Input your answer correct to three decimal places.  \$$x = \$$ [[1]]

", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "scripts": {}, "variableReplacements": [], "maxValue": "{b1}", "mustBeReduced": false, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "allowFractions": false, "minValue": "{b1}", "marks": 1, "strictPrecision": false, "precisionType": "dp", "showPrecisionHint": false, "showFeedbackIcon": true, "precision": "3"}, {"notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "scripts": {}, "variableReplacements": [], "maxValue": "{c1}", "mustBeReduced": false, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "allowFractions": false, "minValue": "{c1}", "marks": 1, "strictPrecision": false, "precisionType": "dp", "showPrecisionHint": false, "showFeedbackIcon": true, "precision": "3"}]}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Solving quadratic equations using a formula,

"}, "variablesTest": {"condition": "", "maxRuns": "1"}, "advice": "

The formula for solving a quadratic equation of the form  \$$ax^2+bx+c=0\$$  is given by

\n

\$$x=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}\$$

\n

In this example  \$$a=\\var{a1},\\,\\,\\,b=\\simplify{+-{a1}*({b1}+{c1})}\$$  and  \$$c=\\simplify{{a1}*{b1}*{c1}}\$$

\n

\$$x=\\frac{\\var{b}\\pm \\sqrt{(-\\var{b})^2-4*\\var{a1}*\\var{c}}}{2*\\var{a1}}\$$

\n

\$$x=\\frac{\\var{b}\\pm \\sqrt{\\simplify{{b}^2-4*{a1}*{c}}}}{\\simplify{2*{a1}}}\$$

\n

\$$x=\\simplify{{b}+({b}^2-4*{a1}*{c})^0.5}/\\simplify{2*{a1}}=\\simplify{({b}+({b}^2-4*{a1}*{c})^0.5)/(2*{a1})}\$$        or        \$$x=\\simplify{{b}-({b}^2-4*{a1}*{c})^0.5}/\\simplify{2*{a1}}=\\simplify{({b}-({b}^2-4*{a1}*{c})^0.5)/(2*{a1})}\$$

\n

\n

", "ungrouped_variables": ["a1", "b1", "c1", "b", "c"], "preamble": {"css": "", "js": ""}, "variables": {"b": {"name": "b", "templateType": "anything", "definition": "{a1}*({b1}+{c1})", "description": "", "group": "Ungrouped variables"}, "a1": {"name": "a1", "templateType": "randrange", "definition": "random(1..6#1)", "description": "", "group": "Ungrouped variables"}, "c": {"name": "c", "templateType": "anything", "definition": "{a1}*{b1}*{c1}", "description": "", "group": "Ungrouped variables"}, "b1": {"name": "b1", "templateType": "randrange", "definition": "random(11..25#1)", "description": "", "group": "Ungrouped variables"}, "c1": {"name": "c1", "templateType": "randrange", "definition": "random(1..10#1)", "description": "", "group": "Ungrouped variables"}}, "type": "question"}, {"name": "Solving quadratic equations 1(b)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "parts": [{"scripts": {}, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "

Type in the greater of the two values that satisfies the equation. Input your answer correct to three decimal places.

\n

\$$x\$$ = [[0]]

\n

Type in the lesser of the two values that satisfies the equation. Input your answer correct to three decimal places.

\n

\$$x\$$ = [[1]]

", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "-{b1}/(2*{a1})+sqrt({b1}^2-4*{a1}*{c1})/(2*{a1})", "mustBeReduced": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "allowFractions": false, "strictPrecision": false, "marks": 1, "variableReplacements": [], "precisionType": "dp", "showPrecisionHint": false, "minValue": "-{b1}/(2*{a1})+sqrt({b1}^2-4*{a1}*{c1})/(2*{a1})", "notationStyles": ["plain", "en", "si-en"], "precision": "3"}, {"precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "-{b1}/(2*{a1})-sqrt({b1}^2-4*{a1}*{c1})/(2*{a1})", "mustBeReduced": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "allowFractions": false, "strictPrecision": false, "marks": 1, "variableReplacements": [], "precisionType": "dp", "showPrecisionHint": false, "minValue": "-{b1}/(2*{a1})-sqrt({b1}^2-4*{a1}*{c1})/(2*{a1})", "notationStyles": ["plain", "en", "si-en"], "precision": "3"}]}], "statement": "

There are two values that satisfy the quadratic function below when  \$$y=\\var{c1}\$$:

\n

\$$y=\\var{a1}x^2+\\var{b1}x\$$

", "rulesets": {}, "variable_groups": [], "ungrouped_variables": ["a1", "b1", "c1"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Solving quadratic equations using a formula,

"}, "variablesTest": {"condition": "b1^2>4*a1*c1", "maxRuns": "1"}, "advice": "

The formula for solving a quadratic equation of the form  \$$ax^2+bx+c=0\$$  is given by

\n

\$$x=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}\$$

\n

In this example  \$$a=\\var{a1},\\,\\,\\,b=\\var{b1}\$$  and  \$$c=\\var{c1}\$$

\n

\$$x=\\frac{-\\var{b1}\\pm \\sqrt{\\var{b1}^2-4\\times\\var{a1}\\times\\var{c1}}}{2\\times\\var{a1}}\$$

\n

\$$x=\\frac{-\\var{b1}\\pm \\sqrt{\\simplify{{b1}^2-4*{a1}*{c1}}}}{\\simplify{2*{a1}}}\$$

\n

\$$x=\\simplify{(-{b1}+ ({b1}^2-4*{a1}*{c1})^0.5)/(2*{a1})}\$$   or   \$$x=\\simplify{(-{b1}- ({b1}^2-4*{a1}*{c1})^0.5)/(2*{a1})}\$$

", "preamble": {"css": "", "js": ""}, "variables": {"a1": {"name": "a1", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(1..6#1)"}, "b1": {"name": "b1", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(16..25#1)"}, "c1": {"name": "c1", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(1..10#1)"}}, "tags": [], "type": "question"}, {"name": "Solving quadratic equations 1(c)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "rulesets": {}, "functions": {}, "type": "question", "preamble": {"css": "", "js": ""}, "statement": "

The following equation can be converted into a quadratic equation:

\n

\$$\\var{a1}x+\\frac{\\simplify{{a1}*{b1}*{c1}}}{x}=\\simplify{{a1}*({b1}+{c1})}\$$

", "showQuestionGroupNames": false, "tags": [], "variable_groups": [], "variablesTest": {"maxRuns": "1", "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Solving quadratic equations using a formula

\$$\\var{a1}x+\\frac{\\simplify{{a1}*{b1}*{c1}}}{x}=\\simplify{{a1}*({b1}+{c1})}\$$

\n

We clear the fraction in the equation by multiplying across by \$$x\$$

\n

\$$\\var{a1}x^2+\\simplify{{a1}*{b1}*{c1}}=\\simplify{{a1}*({b1}+{c1})}x\$$

\n

Bringing all the terms to the left hand side and putting them in order of their powers of \$$x\$$ gives

\n

\$$\\var{a1}x^2-\\simplify{{a1}*({b1}+{c1})}x+\\simplify{{a1}*{b1}*{c1}}=0\$$

\n

The formula for solving a quadratic equation of the form  \$$ax^2+bx+c=0\$$  is given by

\n

\$$x=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}\$$

\n

In this example  \$$a=\\var{a1},\\,\\,\\,b=\\simplify{+-{a1}*({b1}+{c1})}\$$  and  \$$c=\\simplify{{a1}*{b1}*{c1}}\$$

\n

\$$x=\\frac{\\var{b}\\pm \\sqrt{(-\\var{b})^2-4*\\var{a1}*\\var{c}}}{2*\\var{a1}}\$$

\n

\$$x=\\frac{\\var{b}\\pm \\sqrt{\\simplify{{b}^2-4*{a1}*{c}}}}{\\simplify{2*{a1}}}\$$

\n

\$$x=\\simplify{{b}+({b}^2-4*{a1}*{c})^0.5}/\\simplify{2*{a1}}=\\simplify{({b}+({b}^2-4*{a1}*{c})^0.5)/(2*{a1})}\$$        or        \$$x=\\simplify{{b}-({b}^2-4*{a1}*{c})^0.5}/\\simplify{2*{a1}}=\\simplify{({b}-({b}^2-4*{a1}*{c})^0.5)/(2*{a1})}\$$

\n

\n

", "ungrouped_variables": ["a1", "b1", "c1", "b", "c"], "variables": {"a1": {"templateType": "randrange", "definition": "random(1..6#1)", "description": "", "name": "a1", "group": "Ungrouped variables"}, "c1": {"templateType": "randrange", "definition": "random(1..10#1)", "description": "", "name": "c1", "group": "Ungrouped variables"}, "b": {"templateType": "anything", "definition": "{a1}*({b1}+{c1})", "description": "", "name": "b", "group": "Ungrouped variables"}, "b1": {"templateType": "randrange", "definition": "random(11..25#1)", "description": "", "name": "b1", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "definition": "{a1}*{b1}*{c1}", "description": "", "name": "c", "group": "Ungrouped variables"}}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "parts": [{"showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "type": "gapfill", "prompt": "

Type in the greater of the two values that satisfies the equation.

\n

\$$x = \$$ [[0]]

\n

Type in the lesser of the two values that satisfies the equation.

\n

\$$x = \$$ [[1]]

", "marks": 0, "gaps": [{"variableReplacements": [], "marks": 1, "scripts": {}, "allowFractions": false, "showCorrectAnswer": true, "minValue": "{b1}", "strictPrecision": false, "type": "numberentry", "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "{b1}", "precisionPartialCredit": 0, "precision": "3", "showPrecisionHint": true, "precisionType": "dp", "variableReplacementStrategy": "originalfirst"}, {"variableReplacements": [], "marks": 1, "scripts": {}, "allowFractions": false, "showCorrectAnswer": true, "minValue": "{c1}", "strictPrecision": false, "type": "numberentry", "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "{c1}", "precisionPartialCredit": 0, "precision": "3", "showPrecisionHint": true, "precisionType": "dp", "variableReplacementStrategy": "originalfirst"}], "variableReplacementStrategy": "originalfirst"}]}, {"name": "Solving two linear equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "parts": [{"scripts": {}, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "

Input the value for \$$x\$$ as an exact fraction.

\n

\$$x = \$$ [[0]]

\n

Input the value for \$$y\$$ as an exact fraction.

\n

\$$y = \$$ [[1]]

", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "({d}*{r1}-{b}*{r2})/({a}*{d}-{b}*{c})", "mustBeReduced": false, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": true, "allowFractions": true, "minValue": "({d}*{r1}-{b}*{r2})/({a}*{d}-{b}*{c})", "marks": 1, "variableReplacements": [], "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "(-{c}*{r1}+{a}*{r2})/({a}*{d}-{b}*{c})", "mustBeReduced": false, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": true, "allowFractions": true, "minValue": "(-{c}*{r1}+{a}*{r2})/({a}*{d}-{b}*{c})", "marks": 1, "variableReplacements": [], "showFeedbackIcon": true}]}], "statement": "

Solve the following system of simultaneous equations:

\n

\$$\\var{a}x+\\var{b}y=\\var{r1}\$$

\n

and

\n

\$$\\var{c}x+\\var{d}y=\\var{r2}\$$

", "rulesets": {}, "variable_groups": [], "ungrouped_variables": ["a", "b", "c", "d", "r1", "r2"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Solving two simultaneous linear equations

"}, "variablesTest": {"condition": "{a}*{d}>{b}*{c}\n", "maxRuns": 100}, "advice": "

equation (i)    \$$\\var{a}x+\\var{b}y=\\var{r1}\$$

\n

equation (ii)    \$$\\var{c}x+\\var{d}y=\\var{r2}\$$

\n

If we decide to eliminate the \$$x\$$ variables we need to have the same number of \$$x\$$ in both equations

\n

\$$\\var{c}\$$*equation (i)      \$$\\simplify{{c}*{a}}x+\\simplify{{c}*{b}}y=\\simplify{{c}*{r1}}\$$

\n

\$$\\var{a}\$$*equation (ii)     \$$\\simplify{{c}*{a}}x+\\simplify{{d}*{a}}y=\\simplify{{a}*{r2}}\$$

\n

Subtracting gives:

\n

\$$\\simplify{{c}*{b}-{d}*{a}}y=\\simplify{{c}*{r1}-{a}*{r2}}\$$

\n

\$$y=\\simplify{({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})}\$$

\n

Substituting this solution for \$$y\$$ into equation (i) gives

\n

\$$\\var{a}x+\\var{b}*(\\simplify{({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})})=\\var{r1}\$$

\n

\$$\\var{a}x=\\var{r1}-\\var{b}*(\\simplify{({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})})\$$

\n

\$$\\var{a}x=\\simplify{{r1}-{b}*({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})}\$$

\n

\n

\$$x=\\simplify{({r1}-{b}*({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a}))/{a}}\$$

\n

", "preamble": {"css": "", "js": ""}, "variables": {"b": {"name": "b", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..12#1)"}, "d": {"name": "d", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..12#1)"}, "r1": {"name": "r1", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(10..40#1)"}, "c": {"name": "c", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..11#1)"}, "a": {"name": "a", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(1..10#1)"}, "r2": {"name": "r2", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(16..50#1)"}}, "tags": [], "type": "question"}, {"name": "Solve a system of three simultaneous linear equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "tags": [], "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a1", "b1", "c1", "r1", "r2", "r3"], "variables": {"b1": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(2..10#1)", "name": "b1"}, "r1": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(20..42#1)", "name": "r1"}, "a1": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(2..8#1)", "name": "a1"}, "c1": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(3..12#1)", "name": "c1"}, "r3": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(30..60#1)", "name": "r3"}, "r2": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(18..50#1)", "name": "r2"}}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Solve a system of three simultaneous linear equations

"}, "parts": [{"type": "gapfill", "prompt": "

Input the value of \$$x\$$ that satisfies the three equations.

\n

\$$x = \$$ [[0]]

\n

Input the value of \$$y\$$ that satisfies the three equations.

\n

\$$y = \$$ [[1]]

\n

Input the value of \$$z\$$ that satisfies the three equations.

\n

\$$z = \$$ [[2]]

", "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "gaps": [{"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "minValue": "((18-{b1}*{c1})*(3*{r1}-{r3})+({c1}-12)*(6*{r2}-{b1}*{r3}))/((3*{a1}-5)*(18-{b1}*{c1})-(12-5*{b1})*(12-{c1}))", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precision": "3", "showCorrectAnswer": true, "mustBeReducedPC": 0, "allowFractions": false, "strictPrecision": false, "showFeedbackIcon": true, "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "((18-{b1}*{c1})*(3*{r1}-{r3})+({c1}-12)*(6*{r2}-{b1}*{r3}))/((3*{a1}-5)*(18-{b1}*{c1})-(12-5*{b1})*(12-{c1}))", "mustBeReduced": false, "showPrecisionHint": true, "marks": 1, "precisionType": "dp", "correctAnswerStyle": "plain"}, {"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "minValue": "((15-2*{c1})*(2*{r1}-{a1}*{r2})+(3*{a1}-8)*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precision": "3", "showCorrectAnswer": true, "mustBeReducedPC": 0, "allowFractions": false, "strictPrecision": false, "showFeedbackIcon": true, "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "((15-2*{c1})*(2*{r1}-{a1}*{r2})+(3*{a1}-8)*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))", "mustBeReduced": false, "showPrecisionHint": true, "marks": 1, "precisionType": "dp", "correctAnswerStyle": "plain"}, {"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "minValue": "((12-5*{b1})*(2*{r1}-{a1}*{r2})+(4-{b1}*{a1})*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precision": "3", "showCorrectAnswer": true, "mustBeReducedPC": 0, "allowFractions": false, "strictPrecision": false, "showFeedbackIcon": true, "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "((12-5*{b1})*(2*{r1}-{a1}*{r2})+(4-{b1}*{a1})*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))", "mustBeReduced": false, "showPrecisionHint": true, "marks": 1, "precisionType": "dp", "correctAnswerStyle": "plain"}], "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": []}], "functions": {}, "advice": "

(i)    \$$\\var{a1}x+2y+4z=\\var{r1}\$$

\n

(ii)   \$$2x+\\var{b1}y+3z=\\var{r2}\$$

\n

(iii)  \$$5x+6y+\\var{c1}z=\\var{r3}\$$

\n

First reduce the three equations in three unknowns to a two equations in two unknowns problem by eliminating one of the variables.

\n

We can eliminate \$$x\$$ using equations (i) and (ii)

\n

2*(i)     \$$\\simplify{2*{a1}}x+4y+8z=\\simplify{2*{r1}}\$$

\n

\$$\\var{a1}\$$*(ii)    \$$\\simplify{2*{a1}}x+\\simplify{{a1}*{b1}}y+\\simplify{3*{a1}}z=\\simplify{{a1}*{r2}}\$$

\n

Subtracting gives us a new equation

\n

(iv)    \$$\\simplify{(4-{a1}{b1})y+(8-3*{a1})z}=\\simplify{2*{r1}-{a1}*{r2}}\$$

\n

We can also eliminate \$$x\$$ using equations (ii) and (iii)

\n

5*(ii)    \$$10x +\\simplify{5*{b1}}y+15z=\\simplify{5*{r2}}\$$

\n

2*(iii)   \$$10x+12y+\\simplify{2*{c1}}z=\\simplify{2*{r3}}\$$

\n

Subtracting gives us another new equation

\n

(v)     \$$\\simplify{(5*{b1}-12)y+(15-2*{c1})z}=\\simplify{5*{r2}-2*{r3}}\$$

\n

We could then eliminate the \$$y\$$ from these two new equations

\n

\$$\\simplify{5*{b1}-12}\$$*(iv)    \$$\\simplify{(5*{b1}-12)*(4-{a1}{b1})y+(5*{b1}-12)*(8-3*{a1})z}=\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})}\$$

\n

\$$\\simplify{4-{a1}{b1}}\$$*(v)    \$$\\simplify{(4-{a1}{b1})*(5*{b1}-12)y+(4-{a1}{b1})*(15-2*{c1})z}=\\simplify{(4-{a1}{b1})*(5*{r2}-2*{r3})}\$$

\n

Subtracting gives us

\n

\$$\\simplify{(5*{b1}-12)*(8-3*{a1})-(4-{a1}{b1})*(15-2*{c1})}z=\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}{b1})*(5*{r2}-2*{r3})}\$$

\n

Thus

\n

\$$z=\\frac{\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}{b1})*(5*{r2}-2*{r3})}}{\\simplify{(5*{b1}-12)*(8-3*{a1})-(4-{a1}{b1})*(15-2*{c1})}}=\\simplify{decimal{((5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}*{b1})*(5*{r2}-2*{r3}))/( (5*{b1}-12)*(8-3*{a1})-(4-{a1}*{b1})*(15-2*{c1}))}}\$$

\n

We can now back substitute this value for \$$z\$$ into equation (iv) to find the correct value for \$$y\$$ and then back substitute both these values into equation (i) to calculate \$$x\$$.

\n

", "statement": "

Solve the following system of three simultaneous linear equations:

\n

\$$\\var{a1}x+2y+4z=\\var{r1}\$$

\n

and

\n

\$$2x+\\var{b1}y+3z=\\var{r2}\$$

\n

and

\n

\$$5x+6y+\\var{c1}z=\\var{r3}\$$

", "type": "question"}, {"name": "Solving a Linear and a Non-linear system of equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "metadata": {"description": "

Solving a Linear and a Non-linear system of equations

", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "statement": "

Given two equations:

\n

\$$\\var{a1}x+\\var{b1}y=\\var{r1}\$$

\n

and

\n

\$$\\var{c1}x^2+\\var{d1}y^2=\\var{r2}\$$

\n

There are two solutions for \$$x\$$ that satisfy both of these equations and for each \$$x\$$ value there exists a corresponding \$$y\$$ value that forms a solution pair.

", "preamble": {"js": "", "css": ""}, "functions": {}, "variable_groups": [], "variables": {"a1": {"name": "a1", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "r1": {"name": "r1", "definition": "random(10..20#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "c1": {"name": "c1", "definition": "random(1..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "r2": {"name": "r2", "definition": "random(20..50#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "b1": {"name": "b1", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "d1": {"name": "d1", "definition": "random(2..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}}, "variablesTest": {"condition": "(2*{a1}*{d1}*{r1})^2>4*({b1}^2*{c1}+{d1}*{a1}^2)*({d1}*{r1}^2-{b1}^2*{r2})", "maxRuns": 100}, "advice": "

To solve a system that involves a linear equation and a non-linear equation we must use the substitution method.

\n

\$$\\var{a1}x+\\var{b1}y=\\var{r1}\$$

\n

\$$\\var{c1}x^2+\\var{d1}y^2=\\var{r2}\$$

\n

The first equation is a linear equaton, we use this to write one variable in terms of the other.

\n

For example, we could make y the subject of this equation

\n

\$$y=\\frac{\\var{r1}-\\var{a1}x}{\\var{b1}}\$$

\n

We can then insert this for every \$$y\$$ in the non-linear equation to get

\n

\$$\\var{c1}x^2+\\var{d1}*\\left(\\frac{\\var{r1}-\\var{a1}x}{\\var{b1}}\\right)^2=\\var{r2}\$$

\n

\$$\\var{c1}x^2+\\var{d1}*\\frac{(\\var{r1}-\\var{a1}x)^2}{\\var{b1}^2}=\\var{r2}\$$

\n

Multiplying across by \$$\\simplify{{b1}^2}\$$ gives

\n

\$$\\simplify{{c1}*{b1}^2}x^2+\\var{d1}(\\var{r1}-\\var{a1}x)^2-\\simplify{{r2}*{b1}^2}=0\$$

\n

\$$\\simplify{{c1}*{b1}^2}x^2+\\var{d1}\\left(\\simplify{{r1}^2}-\\simplify{2*{r1}*{a1}}x+\\simplify{{a1}^2}x^2\\right)-\\simplify{{r2}*{b1}^2}=0\$$

\n

Gathering the like terms together gives

\n

\$$\\simplify{({c1}*{b1}^2+{d1}*{a1}^2)x^2-(2*{a1}*{d1}*{r1})x+({d1}*{r1}^2-{r2}*{b1}^2)}=0\$$

\n

\n

Recall the formula for solving a quadratic equation of the form  \$$ax^2+bx+c=0\$$  is given by

\n

\$$x=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}\$$

\n

In this example \$$a = \\simplify{({c1}*{b1}^2+{d1}*{a1}^2)}, b = \\simplify{-(2*{a1}*{d1}*{r1})}\$$ and \$$c = \\simplify{{d1}*{r1}^2-{r2}*{b1}^2}\$$

\n

Once we have each \$$x\$$ value we insert it into   \$$y=\\frac{\\var{r1}-\\var{a1}x}{\\var{b1}}\$$   to find the corresponding \$$y\$$ value.

", "parts": [{"showCorrectAnswer": true, "prompt": "

Input the larger of the two \$$x\$$ values.      \$$x = \$$ [[0]]

\n

Input the \$$y\$$ value that corresponds to the previous answer.     \$$y = \$$ [[1]]

\n

\n

Input the lesser of the two \$$x\$$ values that satisfies both equations.     \$$x = \$$ [[2]]

\n

Input the \$$y\$$ value that corresponds to the previous answer.     \$$y = \$$ [[3]]

You should try every question!

"}, "preventleave": true, "startpassword": "3.1415"}, "timing": {"allowPause": true, "timeout": {"action": "warn", "message": "

Times up!

"}, "timedwarning": {"action": "warn", "message": "

You are going to time out soon...