There’s something warm and comforting about the old, reliable Quadratic Formula (QF). You can plug the parameters of any quadratic equation into it, do a little calculating, and easily come up with the correct answer(s). Simple, right?
No, actually, it’s not so simple. Average math students do not come up with the correct answer(s). There are just too many pitfalls along the way:
- If the original equation is not in standard form, almost anything can happen.
- If b turns out to be negative, two things can go wrong. The minus sign in “–b” can be misinterpreted as subtraction rather than opposite, even though there’s nothing to subtract it from. And squaring it can produce a negative number because students type “–62” on their calculator rather than “(–6)2”.
- If a or c turns out to be negative, many students forget that “–4ac” amounts to subtracting a negative and therefore adding a positive. If both turn out to be negative, it’s even worse!
- Fortunately, the “±” sign reminds almost everyone that there are two solutions…except when there aren’t. “Plus or minus the square root of zero” flummoxes too many students. Why is there only one answer?????
- And then there’s the order of operations within the discriminant. A shaky student may calculate b2–4 before multiplying by ac, or make some similar mistake.
- Finally, carelessness about the length of the fraction bar leads to all kinds of issues, such as thinking it’s only the radical expression that’s divided by 2a, not the whole numerator.
You’d think that the algorithmic process of blindly following a formula would be straightforward, but it’s not so.
In addition to the errors above — which can usually be caught by a student who is both careful and confident, but so many people are neither — there are the more abstract issues. Two of them particularly grate on me. One is that only honors students (and not even very many of them) have the faintest idea why this formula works. It’s just plug and chug. At least there’s a chance that each step in other solution methods will make sense, but the QF is just magic. I’ve attempted to derive it, or prove it, for college-prep classes, but their eyes glaze over before you can count to 3. It has to remain magic.
My other issue is its limited applicability. As the name suggests, it works only on quadratic equations. Yes, it works on all quadratics — at least after they’re put into standard form — but it can’t be used on other equations, not even other polynomials. Admittedly, backtracking and factoring have limited applicability as well, not even working on all quadratics, but at least they can be used for certain non-quadratic equations. No one in their right mind would use the cubic formula, and the quartic formula is flatly out of the question, but backtracking and factoring sometimes work for cubics and quartics, showing more general uses for these techniques than the QF has.
Despite all this, we have to teach the Quadratic Formula to all students. Yes, I mean all. Grumble, grumble.