by RonPurewal Wed Sep 17, 2008 4:51 am
the rule is: take the coefficient of x, divide it in half, and then square the resulting number.
here the coefficient of x is -9 (although you can ignore the negative sign, since you're eventually going to square the number anyway - the complete-the-square term will turn out positive no matter what).
cutting that in half yields -9/2
squaring yields 81/4
then you have x^2 - 9x + 81/4, which will factor to (x - 9/2)^2. note that this factorization would be difficult to find at random, but you aren't finding it at random; you already know ahead of time that the expression will factor into a perfect square (hence the expression "completing the square"), and therefore the factorization is easy to find.