Quadratic equation in the complex numbers

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Ulrich Görtz

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Ulrich Görtz 4 years, 4 months ago

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Ulrich Görtz 4 years, 4 months ago

Created this as a copy of Quadratische Gleichung in den komplexen Zahlen.

Name	Status	Author	Last Modified
Quadratische Gleichung in den komplexen Zahlen	Ready to use	Ulrich Görtz	22/11/2020 11:28
Quadratic equation in the complex numbers	draft	Ulrich Görtz	20/11/2020 21:07

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Question statement

Lösen Sie die folgenden quadratischen Gleichungen in den komplexen Zahlen. Die Gleichungen sind so ausgewählt, dass Sie die Lösungen in der Form $a+bi$ für Bruchzahlen (bzw. ganze Zahlen) $a$ und $b$ angeben können.

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			Name	Type	Generated Value

p	decimal	dec("-2")
q	decimal	dec("2")
c1	decimal	dec("1") - i
c2	decimal	dec("1") + i
c	rational	1
d	rational	1

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-c1-c2

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dec("-2")

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Gegeben ist die Gleichung $\simplify[all, fractionNumbers]{x^2 + {p}*x + {q}} = 0$ .

Die Lösungen sind Gap 0. und Gap 1..

Advice

Sie können diese quadratischen Gleichungen wie Gleichungen über den reellen Zahlen behandeln, nur dass Sie die Wurzel aus einer negativen Zahl ziehen müssen. Das ist in den komplexen Zahlen möglich: $\sqrt{-a} = i\sqrt{a}$ (für $a\in\mathbb R_{> 0}$ ).

Im ersten Fall schreiben Sie die Gleichung um als $\simplify[fractionNumbers]{(x+{p/2})^2 - {abs(p)/2}^2 + {q}}$ , also sind die Lösungen $\var[fractionnumbers]{-p/2} \pm i\sqrt{\var[fractionNumbers]{-(p/2)^2+q}}$ , also $\var[fractionNumbers]{c1}$ und $\var[fractionNumbers]{c2}$ .

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Ulrich Görtz 4 years, 4 months ago

Ulrich Görtz 4 years, 4 months ago

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Quadratic equation in the complex numbers

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Ulrich Görtz 4 years, 4 months ago

Ulrich Görtz 4 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: decimal

Parts

Gap-fill

Generated value: `decimal`