We learn to factor polynomials so we can solve for x. Stuff like

3x^{2} + 2x – 1

can be easily factored into

(3x – 1)(x + 1)

But some things aren’t so easy. And some things are just down right a pain in the bottom. Like this one:

18x^{2} + 189x + 490

“Just shoot me,” you might be thinking. But cool your jets, cowboy. Here’s the first tool you need to never factor polynomials again:

I know – who wants to memorize that formula? But wait. If you use that one formula that you can memorize, you never have to factor polynomials again. Watch:

So the factoring isn’t bad on that one, rigth? How about this one:

And how about if you come across one of these. Wouldn’t it be nice to get rid of all guesswork:

This has some disadvantages, of course. You see that there’s a lot more arithmetic. And there’s some things to look out for. But if you hate trial and error (like I do), then you might be willing to take the bad with that good.

Whatcha think?

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The three examples you use are actually all easily factorisable. What about something like 9x^2 +12x – 1 which is not easily factorised?

Anyway, why does this formula work

(I know but does your student)?

Thanks, M, for your comment.

The point of the tip is that you can factor by using the quadratic formula. So that no matter what – you don’t even have to

tryto factor.Using the quadratic formula on 9x^2 + 12x – 1 would be totally fine. Again, if you default to the quad formula, then if they came out as nice numbers, you could show it as factors. If they came out as messy numbers, then you could just finalize your problem by announcing your solution (or zeros, or whatever you were looking for with it).

Now for the question of why. It depends on the age/maturity level of your kids. And depends on if they are top down learners or bottom up learners.

You can show them this and then they can investigate the why. Or if they need to be shown the why, you can derive the quad formula and show the comparison with the zeros on the graph.

This is a tool. It won’t be the best tool for all students, but for some it will be just the thing they need.

Thanks, M, for your comment and question!