Question 164733: Factor 10x^2-21x-10
Found 2 solutions by edjones, gonzo:
Answer by edjones(8007)   (Show Source):
You can put this solution on YOUR website! 10x^2-21x-10
10*-10=-100. What 2 factors of -100 have a sum of -21? Ans -25 and 4.
10x^2+4x-25x-10
=2x(5x+2)-5(5x+2)
=(2x-5)(5x+2)
.
Ed
Answer by gonzo(654)  (Show Source):
You can put this solution on YOUR website! answer is:
(2x-5) * (5x+2) = 0
multiplying out using FOIL, you get
First = 2x * 5x = 10x
Outer = 2x * 2 = 4x
Inner = -5 * 5x = -25x
Last = -5 * 2 = -10
adding all up together, you get
10x^2 - 21x - 10
which is your original equation.
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original equation is 10x^2 - 21x - 10
a = 10
b = -21
c = -10
b^2 - 4ac = 841 which has an integral square root (29) which indicates the roots are integers.
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since you know that the roots are integers, then you need to find the right combination of factors of a and c which will provide the answer.
a = 10 has factors of 10 & 1, or 5 & 2.
c = -10 has factors of 10 & 1, or 5 & 2 also if you neglect the signs for now.
you do cross multiplies and adds to see if the middle factor of b will be correct (ax^2 + bx + c).
using 10 & 1 for a, and 10 & 1 for c, we get 10*10 & 1*1 not equal to -21
we also get 10*1 & 10*1 still not equal to -21.
an a factor of 10 & 1, with a c factor of 10 & 1 is clearly no good.
same with 10 & 1 for a, and 5 & 2 for c.
you get 10*5 & 1*2, or 10*2 & 5*1, neither of which will add or subtract to equal -21.
5 & 2 for a, and 5 & 2 for c, come out as follows:
5*2 & 2*5 give 10 & 10 which don't add or subtract to be = to -21.
5*5 & 2*2 give 25 & 4 which DO equal to -21 when 2*2 is positive and 5*5 is negative.
so the a factors are 5 and 2, and the c factors are 5 and 2, in a specific combination that cross multiplies to be - (5*5) + (2*2).
this comes out to be (5x+2) * (2x-5).
using FOIL to multiply these factors together gets
First (F) = 5x*2x = 10x^2
Outer (O) = 5x * -5 = -25x
Inner (I) = 2*2x = +4x
Last (L) = 2*-5 = -10
this was already done up above and shown to be correct.
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i did use the quadratic equation but didn't come up with the answer directly. not sure why. something for me to research.
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hope this is good enough for what you need.
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