// Numbas version: exam_results_page_options {"name": "Ugur's copy of Ugur's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Ugur's copy of Ugur's copy of Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$", "tags": [], "metadata": {"description": "

Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.

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

Use the quadratic formula to calculate values for $x$ in these equations. Input the possible values as $x_1$ and $x_2$, where $x_1<x_2$.

Round your answers to two decimals. E.g. if you found 4.1567, round it up to 4.16; and if you found 6.653 round it down to 6.65.

", "advice": "

The quadratic formula is

\\[x={\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\\]

a)

We first rearrange our equation into the form $ax^2+bx+c=0$:

\\[\\begin{align}
\\simplify{{b1}x^2+{b2}x+{b3}}&=0=\\var{b4}x\\\\
\\simplify{{b1}x^2+{b2-b4}x+{b3}}&=0\\text{.}
\\end{align}\\]

We can then read off the values for $a, b$ and $c$, which are

\\[\\begin{align}
a&=\\var{b1}\\text{,}\\\\
b&=\\var{b2-b4}\\text{,}\\\\
c&=\\var{b3}\\text{.}
\\end{align}\\]

Substituting these values into the quadratic formula,

\\[x = {\\frac {-\\var{b2-b4}\\pm\\sqrt{\\var{b2-b4}^2-4\\times \\var{b1}\\times \\var{b3}}}{2\\times\\var{b1}}},\\]

we obtain solutions

\\[\\begin{align}
x_1&=\\var{dpformat(p1,2)}\\text{,}\\\\
x_2&=\\var{dpformat(p2,2)}\\text{.}
\\end{align}\\]

", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a3": {"name": "a3", "group": "Ungrouped variables", "definition": "random(-30..4 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(15..20)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(9..a-1)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(2..10 except a1)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "part 2", "definition": "random(-10..4 except 0)^2", "description": "", "templateType": "anything", "can_override": false}, "b4": {"name": "b4", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "(-(b2-b4)+((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "random(-30..6 except a3)", "description": "", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "random(1..15)", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "(-(a2)-((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "(-(b2-b4)-((b2-b4)^2-4*b1*b3)^(1/2))/(2*b1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "part 2", "definition": "2c^0.5+a1^2", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(20..35 except a2)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(9..30 except a1)", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "(-(a2)+((a2)^2-4*a1*(a3-a4))^(1/2))/(2*a1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "a2", "a3", "a4", "b1", "b2", "b3", "b4", "x1", "p1", "p2", "x2", "a", "m"], "variable_groups": [{"name": "part 2", "variables": ["b", "c"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$\\simplify{{b1}x^2+{b2}x+{b3}={b4}x}$

$x_1=$ [[0]]

$x_2=$ [[1]]

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "p1", "maxValue": "p1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "p2", "maxValue": "p2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}]}