- #36
mathwonk
Science Advisor
Homework Helper
- 11,635
- 1,930
It has finally dawned on me how my high school textbook might have made completing the square look more motivated to me. My book had pretty much the derivation in post #27, which made me ask "and how did you think of that?" after the first step. Probably it was my weakness with fractions that was the problem.
I did not readily see that
X^2 + bX + c = (X+b/2)^2 -b^2/4 + c, but I knew very well that (X+b)^2 = X^2 + 2bX + b^2.
So if they had only written it as follows, I would perhaps have found it clearer:
X^2 + 2bX + c = (X^2 + 2bX + b^2) -b^2 + c = (X+b)^2 -b^2 +c. Hence
X^2 + 2bX + c = 0 iff (X+b)^2 = b^2-c iff X+b = ≠ sqrt(b^2-c) iff X = -b ± sqrt(b^2-c).
I.e. X^2 + 2bX almost cries out to be completed by adding b^2.
In the same way, when using the method emphasizing that the cofficients tell us what the sum and product of the roots r,s is, I find it clearer to write X^2 -bX + c, so that b = r+s and c = rs.
I don't know if any of my students would have been helped by this, as although I am still struggling to see how to make things clear, I have long been retired from the classroom. Maybe it would not have helped those for whom the problem was seeing that (X+b)^2 = X^2 + 2bX + b^2, but that could have been explained by the many nice geometric illustrations which have been given here, (going back to Euclid, Book II, Prop. 4)
http://aleph0.clarku.edu/~djoyce/elements/bookII/propII4.htmlWell I just spent a good hour + watching the videos linked in post 22 by Muu9, and in my opinion that guy, James Tanton, has to be the greatest (algebra) teacher in the world. (I was slightly reassured that he is human, when he did the computation -5 -8 = -11 in his head.) Thank you Muu9!
I did not readily see that
X^2 + bX + c = (X+b/2)^2 -b^2/4 + c, but I knew very well that (X+b)^2 = X^2 + 2bX + b^2.
So if they had only written it as follows, I would perhaps have found it clearer:
X^2 + 2bX + c = (X^2 + 2bX + b^2) -b^2 + c = (X+b)^2 -b^2 +c. Hence
X^2 + 2bX + c = 0 iff (X+b)^2 = b^2-c iff X+b = ≠ sqrt(b^2-c) iff X = -b ± sqrt(b^2-c).
I.e. X^2 + 2bX almost cries out to be completed by adding b^2.
In the same way, when using the method emphasizing that the cofficients tell us what the sum and product of the roots r,s is, I find it clearer to write X^2 -bX + c, so that b = r+s and c = rs.
I don't know if any of my students would have been helped by this, as although I am still struggling to see how to make things clear, I have long been retired from the classroom. Maybe it would not have helped those for whom the problem was seeing that (X+b)^2 = X^2 + 2bX + b^2, but that could have been explained by the many nice geometric illustrations which have been given here, (going back to Euclid, Book II, Prop. 4)
http://aleph0.clarku.edu/~djoyce/elements/bookII/propII4.htmlWell I just spent a good hour + watching the videos linked in post 22 by Muu9, and in my opinion that guy, James Tanton, has to be the greatest (algebra) teacher in the world. (I was slightly reassured that he is human, when he did the computation -5 -8 = -11 in his head.) Thank you Muu9!
Last edited: