Solution 2.3:1b
From Förberedande kurs i matematik 1
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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 | ||
| - | {{Displayed math||<math>\ | + | {{Displayed math||<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 | ||
| - | {{Displayed math||<math>x^{2}+2x = \ | + | {{Displayed math||<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 | ||
Current revision
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.
