Solution 2.3:1b
From Förberedande kurs i matematik 1
(Difference between revisions)
m (Robot: Automated text replacement (-[[Bild: +[[Image:)) |
m |
||
(3 intermediate revisions not shown.) | |||
Line 1: | Line 1: | ||
- | {{ | + | 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>\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. | ||
+ | |||
+ | If we use this formula, 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</math>}} | ||
+ | |||
+ | and if we subtract the last "1", we obtain | ||
+ | |||
+ | {{Displayed math||<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, | ||
+ | |||
+ | {{Displayed math||<math>(x+1)^{2}-2 = x^{2}+2x+1-2 = x^{2}+2x-1</math>}} | ||
+ | |||
+ | and see that the relation really holds. |
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.