Question 63231: Factor each polynomial as completely as possible. Please help me understand. thanks!
24b^2-30b-9
Answer by ptaylor(2198)   (Show Source):
You can put this solution on YOUR website! We want to factor 24b^2-30b-9=0
Although this is in quadratic form (Ax^2+Bx+C)=0, I find it much easier to work with (especially when trying to find factors) if the "A" term is 1. When "A" is 1, then the second term is simply the sum of the factors of the third term. So we'll divide both sides by 24 and get:
b^2-(30/24)b-(9/24)=0 lets next reduce the fractions some:
b^2-(10/8)b-(3/8)=0 Thus, our factors are:
(b+?)(b-?)
Now all we need to do is find two factors of -3/8, that when added together gives us -10/8. Well, lets find the factors of -3/8. We'll do this by listing the factors of the numerator and denominator
(a) -3/8=-(3x1)/(4x2)=(-3/4)(+1/2) or (+3/4)(-1/2) or (+or-6/8)(-or+4/8)
(b) -3/8=-(1x3)/(4x2)=(-1/4)(+3/2) or (+1/4)(-3/2) or (+or-2/8)(-or+12/8)
(c) -3/8=-(3x1)/(8x1)=(-3/8)(+1/1) or (+3/8)(-1/1) or (+or-3/8)(-or+8/8)
(d) -3/8=-(3x1)/(1x8)=(-3/1)(+1/8) or (+3/1)(-1/8) or (+or-24/8)(-or+1/8)
I think that's about it for factors. Now which pair adds up to -10/8???
In (b), if we choose +2/8 and a -12/8, we get what we are after. Thus, our factors are:
(b+2/8)(b-12/8)=0 and we can reduce the fractions, getting
(b+1/4)(b-3/2)=0
b=-1/4 and b=3/2
Generally, it's not necessary to go through this level of analysis to find the factors. Many of the possible factors, such as (c) and (d) above, can be quickly eliminated. However, it does involve some trial and error, at times.
Hope this helps some----ptaylor
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