## The Astounding Power of Area

### 3.6 A Comment on Factoring Quadratics

Educators might be surprised that I don’t teach factoring as a part of solving quadratic equations. That is, I don’t teach in a quadratics units how to recognize \(x^{2}-5x+6\) as \((x-2)(x-3)\) and \(6x^{2}-7x-5\) as \((2x+1)(3x-5)\), for example.

I omit factoring from the quadratics unit, and do it later on in the year, for one strong reason.

*Factoring yields expressions that are associated with pictures of rectangles. But solving quadratics is a story of SYMMETRY, using the symmetry of a square to our advantage. (And making full use of symmetry is a powerful technique in mathematics.) Let’s not muddy the story by bringing in a non-symmetrical idea out-of-the-blue.*

The technique we exploit in solving quadratics comes from noticing that we can easily solve a square problem: \(a^2=9\). We cannot, however, solve a rectangle problem \(a\times b = 9\) without extra information. (There are infinitely many solutions to \(ab = 9\).) The symmetry of the square is a potent tool to use.

**WHEN TO TEACH FACTORING**

I’d prefer the answer to be never, except when it is truly needed in a study of Galois Theory.

My concern with the introduction of factoring in a high-school curriculum is that it chiefly introduced as a technique to answer questions that have been designed to be solved by factoring. Most quadratics never factor nicely. (Care to guess the factors of \(x^{2}+x+1\)?).

Because it is a bit of a false construct in the high-school curriculum, I would prefer to bump the study of factoring to a unit on graphing polynomials. (At least this offers a modicum of motivation about why one might care to factor: One can nut one’s way through thinking what the graph of \(p\left(x\right)=\left(x-5\right)^{3}\left(x-1\right)\left(x+2\right)^{51}\left(x+4\right)^{300}\left(x+10\right)\) looks like.)

Of course, moments of needing to factor might occur earlier and I have no trouble with asking students to treat factoring as a logic puzzle about the area model:

*Can you find “nice” numbers that fit this rectangle?*

But keep this work as a separate exercise from a first introduction to solving quadratics. The story of the symmetry of a square needs to well and truly shine through and settle in. Keep factoring very separate, at least at first.

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