I think this is overall an interesting piece, but I think it strays too far off topic. Most of the article deals with the quadratic formula, not with quadratic equations. I think this is an important distinction, and that each topic should have its own page. But before such a major change is enacted, it should probably be discussed.
I also some suggestions for improvement to the article about quadratic equations that I think should be considered.
As usual for a core article, we should really nit-pick to make sure there is a consensus that the first paragraph, and especially the first sentence, is perfect. I suggest changing the word "one" in the first sentence to "equation", and have this link to a page about equations. Through years of teaching, I have found that the typical student does not distinguish between expressions and equations -- linking to equation will allow them to make this distinction.
I also suggest rewriting the rest of the paragraph as follows: "Quadratic equations appear when studying many real-world problems: insert examples here. Being able to solve them is thus one of the fundamental tools for understanding our world with quantitative reasoning. All quadratic equations can be solved using a variety of methods: factoring, completing the square, or by application of the quadratic formula.
The rest of the article
"Using algebra, a quadratic equation can always be put in the form . The solutions, or roots, of a quadratic equation in this form can be found by factoring, completing the square, or through application of the quadratic formula: (insert quadratic formula here). The discriminant determines the number and type of solutions:
- 2 real distinct solutions when b^2-4ac > 0
- 1 real solutions when b^2-4ac = 0
- 2 complex solutions when b^2-4ac < 0
We see that even when a, b, and c are real numbers, the solutions may be complex numbers."
I would then perhaps discuss applications where the solutions are complex, and what this can mean -- perhaps point out that even having a negative root can be troublesome, and that "choosing the positive answer" is often necessary in solving a problem based in a real-world application.
I think the section called "the problem" is very relevant to this article, and should stay -- the same goes for the nice diagrams further down (by the way -- I really like those. What program created them?) But I think the rest of the article should be moved to the quadratic formula page.
Finally, I think it would be nice to put in a little history of quadratic equations -- methods for solving special cases of them and special cases of the quadratic formula were known by various ancient cultures.
Perhaps on an advanced page, one can discuss equations involving quantities other than real numbers -- say quadratic matrix equations, quadratic equations involving p-adic numbers, quadratic equations mod p, discuss applications of these, and when you can identify the number of solutions within the given ring.Barry R. Smith 13:25, 2 May 2008 (CDT)
See advanced talk page
Peter Jackson 18:06, 4 December 2008 (UTC)