Lösung 2.3:2d - Versionsgeschichte

Ombtut3 um 11:56, 9. Sep. 2009

Ombtut3 — Wed, 09 Sep 2009 11:56:10 GMT

← Nächstältere Version		Version vom 11:56, 9. Sep. 2009
Zeile 30:		Zeile 30:

	Also sind die Lösungen richtig.		Also sind die Lösungen richtig.
		+
		+	Alternativer Lösungsweg: [[2.3:2d_alt4\|p-q_Formel]]

Markus.bez um 12:30, 9. Jun. 2009

Markus.bez — Tue, 09 Jun 2009 12:30:31 GMT

← Nächstältere Version		Version vom 12:30, 9. Jun. 2009
Zeile 1:		Zeile 1:
-	Wir normalisieren die Gleichung indem wir alle Terme ~~mit~~ 4 dividieren,	+	Wir normalisieren die Gleichung, indem wir alle Terme durch 4 dividieren:

	{{Abgesetzte Formel\|\|<math>x^{2}-7x+\frac{13}{4}=0\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>x^{2}-7x+\frac{13}{4}=0\,\textrm{.}</math>}}
Zeile 13:		Zeile 13:
	\end{align}</math>}}		\end{align}</math>}}

-	Die Gleichung kann daher ~~wie~~	+	Die Gleichung kann daher als

	{{Abgesetzte Formel\|\|<math>\Bigl(x-\frac{7}{2}\Bigr)^{2} - 9 = 0\,,</math>}}		{{Abgesetzte Formel\|\|<math>\Bigl(x-\frac{7}{2}\Bigr)^{2} - 9 = 0\,,</math>}}
Zeile 23:		Zeile 23:
	:*<math>x-\frac{7}{2}=-\sqrt{9}=-3\,,\quad</math>, also <math>x=\frac{7}{2}-3=\frac{1}{2}.</math>		:*<math>x-\frac{7}{2}=-\sqrt{9}=-3\,,\quad</math>, also <math>x=\frac{7}{2}-3=\frac{1}{2}.</math>

-	Ersetzen ~~wor~~ ''x'' mit 1/2 und 13/2 in der ursprünglichen Gleichung erhalten wir	+	Ersetzen wir ''x'' mit 1/2 und 13/2 in der ursprünglichen Gleichung, erhalten wir

	:*''x'' = 1/2: <math>\ \text{Linke Seite} = 4\cdot\bigl(\tfrac{1}{2}\bigr)^{2} - 28\cdot\tfrac{1}{2}+13 = 4\cdot\tfrac{1}{4}-14+13 = \text{Rechte Seite,}</math>		:*''x'' = 1/2: <math>\ \text{Linke Seite} = 4\cdot\bigl(\tfrac{1}{2}\bigr)^{2} - 28\cdot\tfrac{1}{2}+13 = 4\cdot\tfrac{1}{4}-14+13 = \text{Rechte Seite,}</math>
Zeile 29:		Zeile 29:
	:*''x'' = 13/2: <math>\ \text{Linke Seite} = 4\cdot\bigl(\tfrac{13}{2}\bigr)^{2} - 28\cdot\tfrac{13}{2}+13 = 4\cdot\tfrac{169}{4} - 14\cdot 13 + 13 = \text{Rechte Seite.}</math>		:*''x'' = 13/2: <math>\ \text{Linke Seite} = 4\cdot\bigl(\tfrac{13}{2}\bigr)^{2} - 28\cdot\tfrac{13}{2}+13 = 4\cdot\tfrac{169}{4} - 14\cdot 13 + 13 = \text{Rechte Seite.}</math>

-	~~und also~~ sind die Lösungen richtig.	+	Also sind die Lösungen richtig.

Martin um 17:29, 14. Mär. 2009

Martin — Sat, 14 Mar 2009 17:29:55 GMT

← Nächstältere Version		Version vom 17:29, 14. Mär. 2009
Zeile 1:		Zeile 1:
-	~~The equation can be written in normalized form (i.e. the coefficient in front of ''x''² is 1) by dividing both sides by~~ 4,	+	Wir normalisieren die Gleichung indem wir alle Terme mit 4 dividieren,

	{{Abgesetzte Formel\|\|<math>x^{2}-7x+\frac{13}{4}=0\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>x^{2}-7x+\frac{13}{4}=0\,\textrm{.}</math>}}

-	~~Complete the square on the left-hand side,~~	+	Quadratische Ergänzung ergibt

	{{Abgesetzte Formel\|\|<math>\begin{align}		{{Abgesetzte Formel\|\|<math>\begin{align}
Zeile 13:		Zeile 13:
	\end{align}</math>}}		\end{align}</math>}}

-	~~The equation can therefore be written as~~	+	Die Gleichung kann daher wie

	{{Abgesetzte Formel\|\|<math>\Bigl(x-\frac{7}{2}\Bigr)^{2} - 9 = 0\,,</math>}}		{{Abgesetzte Formel\|\|<math>\Bigl(x-\frac{7}{2}\Bigr)^{2} - 9 = 0\,,</math>}}

-	~~and taking the square root gives the solutions as~~	+	geschrieben werden, und wir erhalten die Wurzeln

-	:*<math>x-\frac{7}{2}=\sqrt{9}=3\,,\quad</math> ~~i.e.~~ <math>x=\frac{7}{2}+3=\frac{13}{2},</math>	+	:*<math>x-\frac{7}{2}=\sqrt{9}=3\,,\quad</math>, also <math>x=\frac{7}{2}+3=\frac{13}{2},</math>

-	:*<math>x-\frac{7}{2}=-\sqrt{9}=-3\,,\quad</math> ~~i.e.~~ <math>x=\frac{7}{2}-3=\frac{1}{2}.</math>	+	:*<math>x-\frac{7}{2}=-\sqrt{9}=-3\,,\quad</math>, also <math>x=\frac{7}{2}-3=\frac{1}{2}.</math>

-	~~As an extra check, we substitute~~ ''x''~~ = ~~1/2 ~~and ''x'' = ~~13/2 ~~into the equation:~~	+	Ersetzen wor ''x'' mit 1/2 und 13/2 in der ursprünglichen Gleichung erhalten wir

-	:*''x'' = 1/2: <math>\ \text{~~LHS~~} = 4\cdot\bigl(\tfrac{1}{2}\bigr)^{2} - 28\cdot\tfrac{1}{2}+13 = 4\cdot\tfrac{1}{4}-14+13 = \text{~~RHS~~,}</math>	+	:*''x'' = 1/2: <math>\ \text{Linke Seite} = 4\cdot\bigl(\tfrac{1}{2}\bigr)^{2} - 28\cdot\tfrac{1}{2}+13 = 4\cdot\tfrac{1}{4}-14+13 = \text{Rechte Seite,}</math>

-	:*''x'' = 13/2: <math>\ \text{~~LHS~~} = 4\cdot\bigl(\tfrac{13}{2}\bigr)^{2} - 28\cdot\tfrac{13}{2}+13 = 4\cdot\tfrac{169}{4} - 14\cdot 13 + 13 = \text{~~RHS~~.}</math>	+	:*''x'' = 13/2: <math>\ \text{Linke Seite} = 4\cdot\bigl(\tfrac{13}{2}\bigr)^{2} - 28\cdot\tfrac{13}{2}+13 = 4\cdot\tfrac{169}{4} - 14\cdot 13 + 13 = \text{Rechte Seite.}</math>
		+
		+	und also sind die Lösungen richtig.

Tekbot: hat „Solution 2.3:2d“ nach „Lösung 2.3:2d“ verschoben: Robot: moved page

Tekbot — Wed, 22 Oct 2008 14:07:25 GMT

hat „Solution 2.3:2d“ nach „Lösung 2.3:2d“ verschoben: Robot: moved page

← Nächstältere Version

Version vom 14:07, 22. Okt. 2008

Tekbot: Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

Tekbot — Wed, 22 Oct 2008 08:32:08 GMT

Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

← Nächstältere Version		Version vom 08:32, 22. Okt. 2008
Zeile 1:		Zeile 1:
	The equation can be written in normalized form (i.e. the coefficient in front of ''x''² is 1) by dividing both sides by 4,		The equation can be written in normalized form (i.e. the coefficient in front of ''x''² is 1) by dividing both sides by 4,

-	{{~~Displayed math~~\|\|<math>x^{2}-7x+\frac{13}{4}=0\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>x^{2}-7x+\frac{13}{4}=0\,\textrm{.}</math>}}

	Complete the square on the left-hand side,		Complete the square on the left-hand side,

-	{{~~Displayed math~~\|\|<math>\begin{align}	+	{{Abgesetzte Formel\|\|<math>\begin{align}
	x^{2}-7x+\frac{13}{4}		x^{2}-7x+\frac{13}{4}
	&= \Bigl(x-\frac{7}{2}\Bigr)^{2} - \Bigl(\frac{7}{2}\Bigr)^{2} + \frac{13}{4}\\[5pt]		&= \Bigl(x-\frac{7}{2}\Bigr)^{2} - \Bigl(\frac{7}{2}\Bigr)^{2} + \frac{13}{4}\\[5pt]
Zeile 15:		Zeile 15:
	The equation can therefore be written as		The equation can therefore be written as

-	{{~~Displayed math~~\|\|<math>\Bigl(x-\frac{7}{2}\Bigr)^{2} - 9 = 0\,,</math>}}	+	{{Abgesetzte Formel\|\|<math>\Bigl(x-\frac{7}{2}\Bigr)^{2} - 9 = 0\,,</math>}}

	and taking the square root gives the solutions as		and taking the square root gives the solutions as

Tek um 07:57, 29. Sep. 2008

Tek — Mon, 29 Sep 2008 07:57:17 GMT

← Nächstältere Version		Version vom 07:57, 29. Sep. 2008
Zeile 25:		Zeile 25:
	As an extra check, we substitute ''x'' = 1/2 and ''x'' = 13/2 into the equation:		As an extra check, we substitute ''x'' = 1/2 and ''x'' = 13/2 into the equation:

-	:*''x'' = 1/2: <math>\ \text{LHS} = 4\cdot\~~Bigl~~(\~~frac~~{1}{2}\~~Bigr~~)^{2} - 28\cdot\~~frac~~{1}{2}+13 = 4\cdot\~~frac~~{1}{4}~~-14+13 = 1~~-14+13 = \text{RHS,}</math>	+	:*''x'' = 1/2: <math>\ \text{LHS} = 4\cdot\bigl(\tfrac{1}{2}\bigr)^{2} - 28\cdot\tfrac{1}{2}+13 = 4\cdot\tfrac{1}{4}-14+13 = \text{RHS,}</math>

-	:*''x'' = 13/2: <math>\ \text{LHS} = 4\cdot\~~Bigl~~(\~~frac~~{13}{2}\~~Bigr~~)^{2} - 28\cdot\~~frac~~{13}{2}+13 = 4\cdot\~~frac~~{169}{4} - 14\cdot 13 ~~+ 13 = 169 - 182~~ + 13 = \text{RHS.}</math>	+	:*''x'' = 13/2: <math>\ \text{LHS} = 4\cdot\bigl(\tfrac{13}{2}\bigr)^{2} - 28\cdot\tfrac{13}{2}+13 = 4\cdot\tfrac{169}{4} - 14\cdot 13 + 13 = \text{RHS.}</math>

Tek um 07:55, 29. Sep. 2008

Tek — Mon, 29 Sep 2008 07:55:16 GMT

← Nächstältere Version		Version vom 07:55, 29. Sep. 2008
Zeile 1:		Zeile 1:
-	The equation can be written in normalized form (i.e. the coefficient in front of	+	The equation can be written in normalized form (i.e. the coefficient in front of ''x''² is 1) by dividing both sides by 4,
-	~~<math>~~x~~^{\text{2}}</math>~~	+
-	is	+
-	~~<math>~~1~~</math>~~	+
-	) by dividing both sides by	+
-	~~<math>~~4~~</math>~~,	+

		+	{{Displayed math\|\|<math>x^{2}-7x+\frac{13}{4}=0\,\textrm{.}</math>}}

-	~~<math>x^{2}~~-~~7x+\frac{13}{4}=0</math>~~	+	Complete the square on the left-hand side,

-		+	{{Displayed math\|\|<math>\begin{align}
-	~~Completing the square on the left-hand side,~~	+	x^{2}-7x+\frac{13}{4}
-		+	&= \Bigl(x-\frac{7}{2}\Bigr)^{2} - \Bigl(\frac{7}{2}\Bigr)^{2} + \frac{13}{4}\\[5pt]
-		+	&= \Bigl(x-\frac{7}{2}\Bigr)^{2} - \frac{49}{4} + \frac{13}{4}\\[5pt]
-	<math>\begin{align}	+	&= \Bigl(x-\frac{7}{2}\Bigr)^{2} - \frac{36}{4}\\[5pt]
-	& x^{2}-7x+\frac{13}{4}=\~~left~~( x-\frac{7}{2} \~~right~~)^{2}-\~~left~~( \frac{7}{2} \~~right~~)^{2}+\frac{13}{4}=\~~left~~( x-\frac{7}{2} \~~right~~)^{2}-\frac{49}{4}+\frac{13}{4} \\	+	&= \Bigl(x-\frac{7}{2}\Bigr)^{2} - 9\,\textrm{.}
-	& =\~~left~~( x-\frac{7}{2} \~~right~~)^{2}-\frac{36}{4}=\~~left~~( x-\frac{7}{2} \~~right~~)^{2}-9 \\	+	\end{align}</math>}}
-	\end{align}</math>	+

	The equation can therefore be written as		The equation can therefore be written as

-		+	{{Displayed math\|\|<math>\Bigl(x-\frac{7}{2}\Bigr)^{2} - 9 = 0\,,</math>}}
-	<math>\~~left~~( x-\frac{7}{2} \~~right~~)^{2}-9=0</math>	+

	and taking the square root gives the solutions as		and taking the square root gives the solutions as

		+	:*<math>x-\frac{7}{2}=\sqrt{9}=3\,,\quad</math> i.e. <math>x=\frac{7}{2}+3=\frac{13}{2},</math>

-	<math>x-\frac{7}{2}=\sqrt{9}=3~~</math>~~	+	:*<math>x-\frac{7}{2}=-\sqrt{9}=-3\,,\quad</math> i.e. <math>x=\frac{7}{2}-3=\frac{1}{2}.</math>
-	~~i.e.~~	+
-	~~<math>x=~~\~~frac{7}{2}+3=\frac{13}{2}~~,~~</math>~~	+
-		+
-		+
-	~~<math>x-~~\~~frac{7}{2}=-\sqrt{9}=-3~~</math>	+
-	i.e.	+
-	<math>x=\frac{7}{2}-3=\frac{1}{2}.</math>	+
-		+
-		+
-	~~As an extra check, we substitute x=1/2 and x=13/2 into the equation:~~	+

		+	As an extra check, we substitute ''x'' = 1/2 and ''x'' = 13/2 into the equation:

-	~~<math>~~x=~~\text{~~1}/~~\text{~~2}</math>: LHS	+	:*''x'' = 1/2: <math>\ \text{LHS} = 4\cdot\Bigl(\frac{1}{2}\Bigr)^{2} - 28\cdot\frac{1}{2}+13 = 4\cdot\frac{1}{4}-14+13 = 1-14+13 = \text{RHS,}</math>
-	~~<math>~~=4\~~centerdot~~ \~~left~~( \frac{1}{2} \~~right~~)^{2}-28\~~centerdot~~ \frac{1}{2}+13=4\~~centerdot~~ \frac{1}{4}-14+13=1-14+13=</math>	+
-	~~RHS~~	+

-	~~<math>~~x=~~\text{~~13}/~~\text{~~2}</math>: LHS	+	:*''x'' = 13/2: <math>\ \text{LHS} = 4\cdot\Bigl(\frac{13}{2}\Bigr)^{2} - 28\cdot\frac{13}{2}+13 = 4\cdot\frac{169}{4} - 14\cdot 13 + 13 = 169 - 182 + 13 = \text{RHS.}</math>
-	~~<math>~~=4\~~centerdot~~ \~~left~~( \frac{13}{2} \~~right~~)^{2}-28\~~centerdot~~ \frac{13}{2}+13=4\~~centerdot~~ \frac{169}{4}-14\~~centerdot~~ 13+13=169-182+13=</math>	+
-	~~RHS~~	+

Ian um 13:53, 20. Sep. 2008

Ian — Sat, 20 Sep 2008 13:53:31 GMT

← Nächstältere Version		Version vom 13:53, 20. Sep. 2008
Zeile 1:		Zeile 1:
-	{{~~NAVCONTENT_START~~}}	+	The equation can be written in normalized form (i.e. the coefficient in front of
-	<~~center~~> ~~[[Image:2_3_2d~~.~~gif]]~~ </~~center~~>	+	<math>x^{\text{2}}</math>
-	{{~~NAVCONTENT_STOP~~}}	+	is
		+	<math>1</math>
		+	) by dividing both sides by
		+	<math>4</math>,
		+
		+
		+	<math>x^{2}-7x+\frac{13}{4}=0</math>
		+
		+
		+	Completing the square on the left-hand side,
		+
		+
		+	<math>\begin{align}
		+	& x^{2}-7x+\frac{13}{4}=\left( x-\frac{7}{2} \right)^{2}-\left( \frac{7}{2} \right)^{2}+\frac{13}{4}=\left( x-\frac{7}{2} \right)^{2}-\frac{49}{4}+\frac{13}{4} \\
		+	& =\left( x-\frac{7}{2} \right)^{2}-\frac{36}{4}=\left( x-\frac{7}{2} \right)^{2}-9 \\
		+	\end{align}</math>
		+
		+	The equation can therefore be written as
		+
		+
		+	<math>\left( x-\frac{7}{2} \right)^{2}-9=0</math>
		+
		+	and taking the square root gives the solutions as
		+
		+
		+	<math>x-\frac{7}{2}=\sqrt{9}=3</math>
		+	i.e.
		+	<math>x=\frac{7}{2}+3=\frac{13}{2},</math>
		+
		+
		+	<math>x-\frac{7}{2}=-\sqrt{9}=-3</math>
		+	i.e.
		+	<math>x=\frac{7}{2}-3=\frac{1}{2}.</math>
		+
		+
		+	As an extra check, we substitute x=1/2 and x=13/2 into the equation:
		+
		+
		+	<math>x=\text{1}/\text{2}</math>: LHS
		+	<math>=4\centerdot \left( \frac{1}{2} \right)^{2}-28\centerdot \frac{1}{2}+13=4\centerdot \frac{1}{4}-14+13=1-14+13=</math>
		+	RHS
		+
		+	<math>x=\text{13}/\text{2}</math>: LHS
		+	<math>=4\centerdot \left( \frac{13}{2} \right)^{2}-28\centerdot \frac{13}{2}+13=4\centerdot \frac{169}{4}-14\centerdot 13+13=169-182+13=</math>
		+	RHS

Tekbot: Lösning 2.3:2d moved to Solution 2.3:2d: Robot: moved page

Tekbot — Tue, 09 Sep 2008 08:57:03 GMT

Lösning 2.3:2d moved to Solution 2.3:2d: Robot: moved page

← Nächstältere Version

Version vom 08:57, 9. Sep. 2008

Tekbot: Robot: Automated text replacement (-[[Bild: +[[Image:)

Tekbot — Thu, 21 Aug 2008 06:45:39 GMT

Robot: Automated text replacement (-[[Bild: +[[Image:)

← Nächstältere Version		Version vom 06:45, 21. Aug. 2008
Zeile 1:		Zeile 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:2_3_2d.gif]] </center>	+	<center> [[Image:2_3_2d.gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}