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Solution 2.3:1c

From Förberedande kurs i matematik 1

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Current revision (14:08, 26 September 2008) (edit) (undo)
 
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<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
<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
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{{Displayed math||<math>x^{2}-ax = \biggl(x-\frac{a}{2}\biggr)^{2} - \biggl(\frac{a}{2}\biggr)^{2}</math>}}
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{{Displayed math||<math>x^{2}-ax = \Bigl(x-\frac{a}{2}\Bigr)^{2} - \Bigl(\frac{a}{2}\Bigr)^{2}</math>}}
and we obtain
and we obtain
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{{Displayed math||<math>x^{2}-2x = \biggl(x-\frac{2}{2}\biggr)^{2} - \biggl(\frac{2}{2}\biggr)^{2} = (x-1)^{2}-1\,\textrm{.}</math>}}
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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\,\textrm{.}</math>}}
This means that
This means that

Current revision

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−1)2−1. 

This means that

5+2x−x2=5−(x2−2x)=5−(x−1)2−1=5−(x−1)2+1=6−(x−1)2.

A quick check shows that we have completed the square correctly

6−(x−1)2=6−(x2−2x+1)=6−x2+2x−1=5+2x−x2.
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