Solution 2.3:2d - Revision history

Tek at 07:57, 29 September 2008

2008-09-29T07:57:17Z

←Older revision		Revision as of 07:57, 29 September 2008
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	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 at 07:55, 29 September 2008

2008-09-29T07:55:16Z

←Older revision		Revision as of 07:55, 29 September 2008
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-	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 at 13:53, 20 September 2008

2008-09-20T13:53:31Z

←Older revision		Revision as of 13:53, 20 September 2008
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-	{{~~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

2008-09-09T08:57:03Z

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

←Older revision

Revision as of 08:57, 9 September 2008

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

2008-08-21T06:45:39Z

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

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2008-04-01T09:58:32Z

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