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Question 473399: I have a factoring issue. I apologize if my question is in the incorrect category.
I am given this trinomial to factor: 14m^2+43m+20.
I understand how to use the "AC method" but do I just sit here and plug numbers into a calculator all day or is there a "shortcut" on how to find the two integers whose product is 280 and whose sum is 43? I just don't "get it", any help is appreciated.
Thanks!
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 14m^2+43m+20
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Check the discriminant to see
that b^2-4ac = 43^2-4*14*20 = 729
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Since it is greater than zero you have 2 unequal Real roots.
And sqrt(729) = 27
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So you have roots at [-43+-27]/(28)
= -16/28 = -4/7 and -70/28 = -10/4 = -5/2
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So the factors of your quadratic are:
(7x+4)(2x+5)
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With the AC Method you were looking for 8 and 35.
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How can you shorten the search.
B = 43 tells you the numbers you want are not around 1,280
sqrt(280) ~ 17 tells you you don't have to look beyond 17
for the smaller of the 2 numbers.
That might lead you to try 10,28; you fall short of 43 but
you are close.
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Another shortcut beside the discriminant is graphing.
Graph the quadratic and y = 0 and see where they intersect.
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Cheers,
Stan H.
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