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		Current revision (14:08, 26 September 2008) (edit) (undo) Tek (Talk | contribs)
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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>, which we also can write as <math>-(x^{2}-2x)</math>. If we neglect the minus sign, we can complete square of the expression <math>2x-x^{2}</math> by using the formula
-	{{NAVCONTENT_STOP}}	+
		+	{{Displayed math||<math>x^{2}-ax = \Bigl(x-\frac{a}{2}\Bigr)^{2} - \Bigl(\frac{a}{2}\Bigr)^{2}</math>}}
		+
		+	and we obtain
		+
		+	{{Displayed math||<math>x^{2}-2x = \Bigl(x-\frac{2}{2}\Bigr)^{2} - \Bigl(\frac{2}{2}\Bigr)^{2} = (x-1)^{2}-1\,\textrm{.}</math>}}
		+
		+	This means that
		+
		+	{{Displayed math||<math>\begin{align}
		+	5+2x-x^{2} &= 5-(x^{2}-2x) = 5-\bigl((x-1)^{2}-1\bigr)\\[5pt]
		+	&= 5-(x-1)^{2}+1 = 6-(x-1)^{2}\textrm{.}
		+	\end{align}</math>}}
		+
		+	A quick check shows that we have completed the square correctly
		+
		+	{{Displayed math||<math>\begin{align}
		+	6-(x-1)^{2}
		+	&= 6-(x^{2}-2x+1)\\[5pt]
		+	&= 6-x^{2}+2x-1\\[5pt]
		+	& =5+2x-x^{2}\textrm{.}
		+	\end{align}</math>}}

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Current revision

As always when completing the square, we focus on the quadratic and linear terms \displaystyle 2x-x^{2}, which we also can write as \displaystyle -(x^{2}-2x). If we neglect the minus sign, we can complete square of the expression \displaystyle 2x-x^{2} by using the formula

\displaystyle x^{2}-ax = \Bigl(x-\frac{a}{2}\Bigr)^{2} - \Bigl(\frac{a}{2}\Bigr)^{2}

and we obtain

\displaystyle x^{2}-2x = \Bigl(x-\frac{2}{2}\Bigr)^{2} - \Bigl(\frac{2}{2}\Bigr)^{2} = (x-1)^{2}-1\,\textrm{.}

This means that

\displaystyle \begin{align} 5+2x-x^{2} &= 5-(x^{2}-2x) = 5-\bigl((x-1)^{2}-1\bigr)\\[5pt] &= 5-(x-1)^{2}+1 = 6-(x-1)^{2}\textrm{.} \end{align}

A quick check shows that we have completed the square correctly

\displaystyle \begin{align} 6-(x-1)^{2} &= 6-(x^{2}-2x+1)\\[5pt] &= 6-x^{2}+2x-1\\[5pt] & =5+2x-x^{2}\textrm{.} \end{align}