Solution 2.1:6d - Revision history

Tek at 11:49, 23 September 2008

2008-09-23T11:49:09Z

←Older revision		Revision as of 11:49, 23 September 2008
Line 1:		Line 1:
	First, we simplify the numerator and denominator for the whole fraction by rewriting as		First, we simplify the numerator and denominator for the whole fraction by rewriting as

-		+	{{Displayed math\|\|<math>\begin{align}
-	<math>\begin{align}	+	a-b+\frac{b^{2}}{a+b}
-	& a-b+\frac{b^{2}}{a+b}=~~\left~~( a-b ~~\right~~)\~~centerdot~~ \frac{a+b}{a+b}+\frac{b^{2}}{a+b}=\frac{~~\left~~( a-b ~~\right~~)\~~centerdot \left~~( a+b ~~\right~~)+b^{2}}{a+b} \\	+	&= (a-b)\cdot\frac{a+b}{a+b} + \frac{b^{2}}{a+b} = \frac{(a-b)\cdot (a+b)+b^{2}}{a+b}\\[5pt]
-	& \frac{a^{2}-b^{2}+b^{2}}{a+b}=\frac{a^{2}}{a+b} \\	+	&= \frac{a^{2}-b^{2}+b^{2}}{a+b} = \frac{a^{2}}{a+b}\,,\\[15pt]
-	\~~end{align}</math>~~	+	1-\biggl(\frac{a-b}{a+b}\biggr)^{2}
-		+	&= 1-\frac{(a-b)^{2}}{(a+b)^{2}} = \frac{(a+b)^{2}}{(a+b)^{2}} - \frac{(a-b)^{2}}{(a+b)^{2}}\\[5pt]
-		+	&= \frac{(a+b)^{2}-(a-b)^{2}}{(a+b)^{2}}\\[5pt]
-		+	&= \frac{(a^{2}+2ab+b^{2})-(a^{2}-2ab+b^{2})}{(a+b)^{2}} = \frac{4ab}{(a+b)^{2}}\,\textrm{.}
-	~~<math>\begin{align}~~	+	\end{align}</math>}}
-	& 1-\~~left~~( \frac{a-b}{a+b} \~~right~~)^{2}=1-\frac{~~\left~~( a-b ~~\right~~)^{2}}{~~\left~~( a+b ~~\right~~)^{2}}=\frac{~~\left~~( a+b ~~\right~~)^{2}}{~~\left~~( a+b ~~\right~~)^{2}}-\frac{~~\left~~( a-b ~~\right~~)^{2}}{~~\left~~( a+b ~~\right~~)^{2}} \\	+
-	& =\frac{~~\left~~( a+b ~~\right~~)^{2}-~~\left~~( a-b ~~\right~~)^{2}}{~~\left~~( a+b ~~\right~~)^{2}} \\	+
-	& =\frac{~~\left~~( a^{2}+2ab+b^{2} ~~\right~~)-~~\left~~( a^{2}-2ab+b^{2} ~~\right~~)}{~~\left~~( a+b ~~\right~~)^{2}}=\frac{4ab}{~~\left~~( a+b ~~\right~~)^{2}} \\	+
-	\end{align}</math>	+
-		+

	The whole fraction is therefore		The whole fraction is therefore

-		+	{{Displayed math\|\|<math>\frac{a-b+\dfrac{b^{2}}{a+b}}{1-\biggl(\dfrac{a-b}{a+b}\biggr)^{2}} = \frac{\dfrac{a^{2}}{a+b}}{\dfrac{4ab}{(a+b)^{2}}} = \frac{a^{2}}{a+b}\cdot\frac{(a+b)^{2}}{4ab} = \frac{a(a+b)}{4b}\,\textrm{.}</math>}}
-	<math>\frac{a-b+\~~frac~~{b^{2}}{a+b}}{1-\~~left~~( \~~frac~~{a-b}{a+b} \~~right~~)^{2}}=\frac{\~~frac~~{a^{2}}{a+b}}{\~~frac~~{4ab}{~~\left~~( a+b ~~\right~~)^{2}}}=\frac{a^{2}}{a+b}\~~centerdot~~ \frac{~~\left~~( a+b ~~\right~~)^{2}}{4ab}=\frac{a~~\left~~( a+b ~~\right~~)}{4b}</math>	+

Ian at 11:23, 16 September 2008

2008-09-16T11:23:08Z

←Older revision		Revision as of 11:23, 16 September 2008
Line 1:		Line 1:
-	{{~~NAVCONTENT_START~~}}	+	First, we simplify the numerator and denominator for the whole fraction by rewriting as
-	<~~center~~> ~~[[Image:2_1_6d.gif]]~~ </~~center~~>	+
-	{{~~NAVCONTENT_STOP~~}}	+
		+	<math>\begin{align}
		+	& a-b+\frac{b^{2}}{a+b}=\left( a-b \right)\centerdot \frac{a+b}{a+b}+\frac{b^{2}}{a+b}=\frac{\left( a-b \right)\centerdot \left( a+b \right)+b^{2}}{a+b} \\
		+	& \frac{a^{2}-b^{2}+b^{2}}{a+b}=\frac{a^{2}}{a+b} \\
		+	\end{align}</math>
		+
		+
		+
		+	<math>\begin{align}
		+	& 1-\left( \frac{a-b}{a+b} \right)^{2}=1-\frac{\left( a-b \right)^{2}}{\left( a+b \right)^{2}}=\frac{\left( a+b \right)^{2}}{\left( a+b \right)^{2}}-\frac{\left( a-b \right)^{2}}{\left( a+b \right)^{2}} \\
		+	& =\frac{\left( a+b \right)^{2}-\left( a-b \right)^{2}}{\left( a+b \right)^{2}} \\
		+	& =\frac{\left( a^{2}+2ab+b^{2} \right)-\left( a^{2}-2ab+b^{2} \right)}{\left( a+b \right)^{2}}=\frac{4ab}{\left( a+b \right)^{2}} \\
		+	\end{align}</math>
		+
		+
		+	The whole fraction is therefore
		+
		+
		+	<math>\frac{a-b+\frac{b^{2}}{a+b}}{1-\left( \frac{a-b}{a+b} \right)^{2}}=\frac{\frac{a^{2}}{a+b}}{\frac{4ab}{\left( a+b \right)^{2}}}=\frac{a^{2}}{a+b}\centerdot \frac{\left( a+b \right)^{2}}{4ab}=\frac{a\left( a+b \right)}{4b}</math>

Tekbot: Lösning 2.1:6d moved to Solution 2.1:6d: Robot: moved page

2008-09-09T08:40:03Z

Lösning 2.1:6d moved to Solution 2.1:6d: Robot: moved page

←Older revision

Revision as of 08:40, 9 September 2008

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

2008-08-21T06:37:09Z

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

←Older revision		Revision as of 06:37, 21 August 2008
Line 1:		Line 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:2_1_6d.gif]] </center>	+	<center> [[Image:2_1_6d.gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}

Lina: Ny sida: {{NAVCONTENT_START}} Bild:2_1_6d.gif {{NAVCONTENT_STOP}}

2008-03-31T11:18:17Z

Ny sida: {{NAVCONTENT_START}} <center> Bild:2_1_6d.gif </center> {{NAVCONTENT_STOP}}

New page

{{NAVCONTENT_START}}
<center> [[Bild:2_1_6d.gif]] </center>
{{NAVCONTENT_STOP}}