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{"name": "Quadratische Gleichung in den komplexen Zahlen", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quadratische Gleichung in den komplexen Zahlen", "tags": [], "metadata": {"description": "<p>Solve two quadratic equations (with real coefficients) in the complex numbers. The solutions have non-zero imaginary part, fractions can appear (but the denominators are rather small).</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>L&ouml;sen Sie die folgenden quadratischen Gleichungen in den komplexen Zahlen. Die Gleichungen sind so ausgew&auml;hlt, dass Sie die L&ouml;sungen in der Form $a+bi$ f&uuml;r Bruchzahlen (bzw. ganze Zahlen) $a$ und $b$ angeben k&ouml;nnen. Geben Sie Ihre Antwort ohne Klammern ein.</p>", "advice": "<p>Sie k&ouml;nnen diese quadratischen Gleichungen wie Gleichungen &uuml;ber den reellen Zahlen behandeln, nur dass Sie die Wurzel aus einer negativen Zahl ziehen m&uuml;ssen. Das ist in den komplexen Zahlen m&ouml;glich: $\\sqrt{-a} = i\\sqrt{a}$ (f&uuml;r $a\\in\\mathbb R_{&gt; 0}$).</p>\n<p>Im ersten Fall schreiben Sie die Gleichung um als $\\simplify[fractionNumbers]{(x+{p/2})^2 - {abs(p)/2}^2 + {q}}$, also sind die L&ouml;sungen $\\var[fractionnumbers]{-p/2} \\pm i\\sqrt{\\var[fractionNumbers]{-(p/2)^2+q}}$, also $\\var[fractionNumbers]{c1}$ und $\\var[fractionNumbers]{c2}$.</p>\n<p>Im zweiten Fall schreiben Sie die Gleichung um als $\\simplify[fractionNumbers]{(x+{pb/2})^2 - {abs(pb)/2}^2 + {qb}}$, also sind die L&ouml;sungen $\\var[fractionnumbers]{-pb/2} \\pm i\\sqrt{\\var[fractionNumbers]{-(pb/2)^2+qb}}$, also $\\var[fractionNumbers]{c1b}$ und $\\var[fractionNumbers]{c2b}$.</p>", "rulesets": {}, "extensions": [], "variables": {"p": {"name": "p", "group": "Ungrouped variables", "definition": "-c1-c2", "description": "", "templateType": "anything"}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "c1*c2", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "c - d*i", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "c+d*i", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-3..7 except 0)/random(1..3)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-3..7 except 0)/random(1..3)", "description": "", "templateType": "anything"}, "cb": {"name": "cb", "group": "Ungrouped variables", "definition": "random(-6..3 except 0)/random(1..4)", "description": "", "templateType": "anything"}, "db": {"name": "db", "group": "Ungrouped variables", "definition": "random(-3..7 except 0)/random(1..3)", "description": "", "templateType": "anything"}, "c1b": {"name": "c1b", "group": "Ungrouped variables", "definition": "cb - i*db", "description": "", "templateType": "anything"}, "c2b": {"name": "c2b", "group": "Ungrouped variables", "definition": "cb+i*db", "description": "", "templateType": "anything"}, "pb": {"name": "pb", "group": "Ungrouped variables", "definition": "-c1b-c2b", "description": "", "templateType": "anything"}, "qb": {"name": "qb", "group": "Ungrouped variables", "definition": "c1b*c2b", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "!(c1=c1b and c2=c2b)", "maxRuns": 100}, "ungrouped_variables": ["p", "q", "c1", "c2", "c", "d", "cb", "db", "c1b", "c2b", "pb", "qb"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// console.log(Numbas.jme.evaluate(this.studentAnswer[0], this.question.scope));\nvalues = [];\nfor(var i=0; 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