For some quadratics such as:
$$10x^2+89x-9$$
I factoring by making the observations $10=(1)(10)=(5)(2)$ and $9=(3)(3)=9(1)$, then I try:
$$(10x ? 1)(x ? 9)$$ $$(10x ? 9)(x ? 1)$$ $$(10 x ? 3)(x ? 3)$$ $$(5x ? 1)(2x ? 10)$$ $$(5x ? 10)(2x ? 1)$$ $$(5x ? 3)(2x ? 3)$$
We got lucky , by our first guess, we can get;
$$(10x-1)(x+9)$$
This is the way my Algebra $2$ teacher made me switch to because he said it's less magic and more factoring. This method is a bit hard to teach, or is it not in your opinion?
But in many cases I find myself going through a lot of possibilities with this method, sometimes to find that my quadratic cannot be factored.
Of course there is the method what multiplies to $-90$ and adds to $89$...this may take a little less thought then we proceed as:
$$10x^2+90x-x-9$$
And of course we may go straight to the quadratic formula...(if the question is to factor then, using the roots we can find a factored form).
If I'm taking an sat or some other timed test, what method would you suggest. And do you have a better method?
How do you proceed with your method? I was taught your teacher's method 30 years ago but have switched to teaching along the lines of your method because it's nearly foolproof and less guess and check. However the current theory in pedagogy is that different methods are acceptable and even beneficial so making you switch methods isn't ideal. As to which is the fastest, I think they are about the same except with lots of practice you can eventually see the factors very quickly by your teacher's method. I don't think the effort practicing that is worthwhile unless it's for competition math though.