From: Pat Ballew (for Teacher2Teacher Service)
Date: Jan 01, 2007 at 16:40:33
Subject: Re: "Bottoms up" as a method to factor trinomials

Well, the secret is that 8 = 24/3...

If you consider that the solutions of x^2 +  bx +c = 0  are the same as the
solutions of 2x^2 + 2bx + 2c  etc... then you are a step closer to
understanding the solution....

If we take 3x^2 + 14x + 8 = 0  and (x=u/3) and substitute we get


(3(u/3)^2 + 14 (u/3)  + 24/3)  = 0  and now if we simplify the first term
we get

u^2/3 + 14 u/3 + 24/3 = 0

now if we multiply all terms by 3 we get

u^2 + 14u + 24... and solve to get the two solutions you had, u=12 and u=2,
but remember that we wanted x, not u, and x=u/3 thus the final solution...

A similar mathod you might have seen that works on the same method involves
multiplying each term by the a coefficient to make it a perfect square...

Using your second example, we rewrite 12x^2 + 13x-35 as

144x^2 + 156 x - 420 = 0   and write factors with 12x in each lead

(12x +    )  (12x -    )  and use the ordinary method, but with larger
numbers to produce the same result....

As a third approach.. you can simply divide everything by the a term...

x^2 + 13/12 x - 35/12 = 0  and then look for two numbers that multiply to
make -35/12 (which of course is -420/144)  and add up to 13/12... if we
assume denominators of 12, then the choice is the same as two numbers that
have a product of -420 and a sum of 13...
Hope that is all clear...
Good luck

 -Pat Ballew, for the T2T service

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