This factored? 8x^2 + 6x - 9 to (4x - 3)(2x + 3)

[itgl72]

How did this factorization come to be?

I see where 4x and 2x came from, and I under stand that -3*3=-9 for the C position, but how did the 6 get there in the B position? Isn't it supposed to be what you get when you add -3 +3? That would be 0, maybe something Im missing?

[maggiemagnet]

I don't understand: What do you mean by "the 6 getting into the 8 position"?

[itgl72]

I don't understand: What do you mean by "the 6 getting into the 8 position"?

Not 8, B, A + B + C, you know...

[maggiemagnet]

Not 8, B, A + B + C, you know...

Oops! OK, I think you're trying to use the a-b-c method for trinomials with a leading coefficient of 1?? To learn how to factor with other leading coefficients, try this lesson. Study "The Hard Case". It'll show that you're wanting numbers that multiply to a*c and that add to b. In your example, that means you need numbers that multiply to 8*-9=-72 and add to +6 (that's how the "6" comes into play), so you'll use +12 and -6.

[LoveMath]

To answer where the B term came from in the expression ax^2+bx+c, are you familiar with the foil method of multipying a binomial by a binomial? If not, it's an acronymn that stands for First, Outer, Inner, Last.

First: multilipy each of the first terms in both parentheses. This gives you (4x)(2x) = 8x^2
Outer: multiply each of the outermost terms in both parentheses. These are: (4x)(+3) = +12x
Inner: multiply each of the innermost terms in both parentheses. These are (-3)(2x) = -6x
Last: multiply each of the last (or second) terms in both parentheses. These are: (-3)(+3) = -9

Then if you combine the outer and inner terms of +12x and -6x you will get +6x.
And this is where the middle term comes from.

You might be able to visualize this concept by putting your hands by the 2 outer terms, multiply them and here we got +12x.
Then place your hands in the center by the two innermost terms, multiply them and you get -6x.
I generally just repeat this process several times to students...outer, inner....then add the two.

Make sense?