Processing Math: Done

From Förberedande kurs i matematik 1

Revision as of 08:55, 9 September 2008 (edit) Tekbot (Talk | contribs) m (Lösning 2.3:1c moved to Solution 2.3:1c: Robot: moved page) ← Previous diff		Revision as of 11:19, 12 September 2008 (edit) (undo) Ian (Talk | contribs) Next diff →
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-	{{NAVCONTENT_START}}	+	As always when completing the square, we focus on the quadratic and linear terms
-	<center> [[Image:2_3_1c.gif]] </center>	+	<math>2x-x^{2}</math>
-	{{NAVCONTENT_STOP}}	+	, which we also can write as
		+	<math>-\left( x^{2}-2x \right)</math>
		+	. If we neglect the minus sign, we can complete square of the expression
		+	<math>2x-x^{2}</math>
		+	by using the formula
		+
		+
		+	<math>x^{2}-ax=\left( x-\frac{a}{2} \right)^{2}-\left( \frac{a}{2} \right)^{2}</math>
		+
		+
		+	and we obtain
		+
		+
		+	<math>x^{2}-2x=\left( x-\frac{2}{2} \right)^{2}-\left( \frac{2}{2} \right)^{2}=\left( x-1 \right)^{2}-1</math>
		+
		+
		+	This means that
		+
		+
		+	<math>\begin{align}
		+	& 5+2x-x^{2}=5-\left( x^{2}-2x \right)=5-\left( \left( x-1 \right)^{2}-1 \right) \\
		+	& \\
		+	& =5-\left( x-1 \right)^{2}+1=6-\left( x-1 \right)^{2} \\
		+	& \\
		+	\end{align}</math>
		+
		+
		+	A quick check shows that we have completed the square correctly.:
		+
		+
		+	<math>\begin{align}
		+	& 6-\left( x-1 \right)^{2}=6-\left( x^{2}-2x+1 \right)=6-x^{2}+2x-1 \\
		+	& \\
		+	& =5+2x-x^{2} \\
		+	& \\
		+	\end{align}</math>

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Revision as of 11:19, 12 September 2008

As always when completing the square, we focus on the quadratic and linear terms 2x−x2 , which we also can write as −x2−2x  . If we neglect the minus sign, we can complete square of the expression 2x−x2 by using the formula

x2−ax=x−2a2−2a2 

and we obtain

x2−2x=x−222−222=x−12−1 

This means that

5+2x−x2=5−x2−2x=5−x−12−1=5−x−12+1=6−x−12

A quick check shows that we have completed the square correctly.:

6−x−12=6−x2−2x+1=6−x2+2x−1=5+2x−x2