How do you factor 12x2 plus 28x-17?

Answer 1

The trick with this kind of question is to find two terms that add up to the middle one (28x), and whose coefficients are equal multiples of the first and last term:

12x2 + 28x - 17

= 12x2 - 6x + 34x - 17

= 6x(2x - 1) + 17(2x - 1)

= (6x + 17)(2x - 1)

The only trick with this technique is that sometimes it's difficult to find those two middle terms (-6x and 34x in this case). If it takes too long to figure this out, you can actually break it down into ~another~ quadratic equation, which will usually be easier to factor, and give you those two values. Consider:

Let "a" and "b" be the unknown coefficients. We know:

b = 28 - a <-- we know this because they must add up to the middle term

12/a = b/17 <-- this must be true in order for us to end up with a common multiple.

∴ 12/a = (28 - a)/17

∴ 28a - a2 = 204

∴ a2 - 28a + 204 = 0

We can then solve for "a" by factoring this equation:

(a - 34)(a + 6) = 0

So our coefficients are the two possible values for "a", 34 and -6. We can then simply plug them in to the original equation and factor it out as shown above.