Lösung 2.3:1b

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When we complete the square, it is only the first two terms, <math>x^{2}+2x</math>, that are involved. The general formula for completing the square states that <math>x^{2}+ax</math> equals
When we complete the square, it is only the first two terms, <math>x^{2}+2x</math>, that are involved. The general formula for completing the square states that <math>x^{2}+ax</math> equals
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{{Displayed math||<math>\Bigl(x+\frac{a}{2}\Bigr)^{2} - \Bigl(\frac{a}{2}\Bigr)^{2}\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<math>\Bigl(x+\frac{a}{2}\Bigr)^{2} - \Bigl(\frac{a}{2}\Bigr)^{2}\,\textrm{.}</math>}}
Note how the coefficient ''a'' in front of the ''x'' turns up halved in two places.
Note how the coefficient ''a'' in front of the ''x'' turns up halved in two places.
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If we use this formula, we obtain
If we use this formula, we obtain
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{{Displayed math||<math>x^{2}+2x = \Bigl(x+\frac{2}{2}\Bigr)^{2} - \Bigl(\frac{2}{2}\Bigr)^{2} = (x+1)^{2}-1</math>}}
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{{Abgesetzte Formel||<math>x^{2}+2x = \Bigl(x+\frac{2}{2}\Bigr)^{2} - \Bigl(\frac{2}{2}\Bigr)^{2} = (x+1)^{2}-1</math>}}
and if we subtract the last "1", we obtain
and if we subtract the last "1", we obtain
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{{Displayed math||<math>x^{2}+2x-1 = (x+1)^{2}-1-1 = (x+1)^{2}-2\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<math>x^{2}+2x-1 = (x+1)^{2}-1-1 = (x+1)^{2}-2\,\textrm{.}</math>}}
To be completely certain that we have used the correct formula, we can expand the quadratic on the right-hand side,
To be completely certain that we have used the correct formula, we can expand the quadratic on the right-hand side,
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{{Displayed math||<math>(x+1)^{2}-2 = x^{2}+2x+1-2 = x^{2}+2x-1</math>}}
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{{Abgesetzte Formel||<math>(x+1)^{2}-2 = x^{2}+2x+1-2 = x^{2}+2x-1</math>}}
and see that the relation really holds.
and see that the relation really holds.

Version vom 08:31, 22. Okt. 2008

When we complete the square, it is only the first two terms, \displaystyle x^{2}+2x, that are involved. The general formula for completing the square states that \displaystyle x^{2}+ax equals

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

Note how the coefficient a in front of the x turns up halved in two places.

If we use this formula, we obtain

\displaystyle x^{2}+2x = \Bigl(x+\frac{2}{2}\Bigr)^{2} - \Bigl(\frac{2}{2}\Bigr)^{2} = (x+1)^{2}-1

and if we subtract the last "1", we obtain

\displaystyle x^{2}+2x-1 = (x+1)^{2}-1-1 = (x+1)^{2}-2\,\textrm{.}

To be completely certain that we have used the correct formula, we can expand the quadratic on the right-hand side,

\displaystyle (x+1)^{2}-2 = x^{2}+2x+1-2 = x^{2}+2x-1

and see that the relation really holds.

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_2.3:1b“