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Thread: factoring

  1. #1
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    factoring

    Can someone help me understand how to factor these questions? I mean work out one of them step by step so, I can try to figure one out. Thanks a bunch!
    1. Factor completely.
    2x2 + 40x + 200
    a) 2(x + 10)(x + 20)
    b) 2(x + 20)2
    c) (2x + 10)2
    d) 2(x + 10)2
    e) none of the above

    2. Factor 4x2 - 20x + 25. Select the answer.
    a) 2x-5
    b) (2x-5)2
    c) (2x+5)2
    d) 2x+5
    e) none of these
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by starrynight View Post
    1. Factor completely.
    2x2 + 40x + 200
    a) 2(x + 10)(x + 20)
    b) 2(x + 20)2
    c) (2x + 10)2
    d) 2(x + 10)2
    e) none of the above
    Well even if you can't manage to factor them you can multiply each of the answers out to see which one is correct.

    $\displaystyle 2x^2 + 40x + 200$

    First note that there is a common factor of 2 in each term:
    $\displaystyle 2x^2 + 40x + 200 = 2(x^2 + 20x + 100)$
    (so you should know automatically that c) can't be correct, right?)

    Now let's take a look at $\displaystyle x^2 + 20x + 100$.

    Take the leading coefficient (1) and multiply it by the constant term (100), in this case giving us 100.

    Now list all possible pairs of factors of 100:
    1, 100
    2, 50
    4, 25
    5, 20
    10, 10
    20, 5
    25, 4
    50, 2
    100, 1
    (and we have all the negatives as well: -1, 100 etc.)

    Now find out which of these pairs add up to be the linear coefficient (20). I get that 10 + 10 = 20.

    So what you want to do is to write 20x = 10x + 10x in your quadratic. So we get
    $\displaystyle 2x^2 + 40x + 200 = 2(x^2 + 20x + 100)$

    $\displaystyle = 2(x^2 + 10x + 10x + 100)$

    $\displaystyle = 2([x^2 + 10x] + [10x + 100])$

    $\displaystyle = 2(x[x + 10] + 10[x + 10])$

    $\displaystyle = 2(x + 10)(x + 10) = 2(x + 10)^2$

    Try this approach on your second problem.

    -Dan
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  3. #3
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    Thanks for your help!
    I think the answer is none of the above. Am I right?
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by starrynight View Post
    Thanks for your help!
    I think the answer is none of the above. Am I right?
    As I said before you can find the correct answer simply by FOILing out each possible answer, so there should be no doubt in your mind.

    No, you don't have it correct.
    $\displaystyle 4x^2 - 20x + 25$

    Using the same method as I posted earlier:
    4 times 25 = 100

    Factors of 100:
    1, 100
    2, 50
    etc.

    And as I said, don't forget about the double negative pairs like -2, -50 etc.

    I get that -20 = -10 + -10, so put the quadratic in the form:
    $\displaystyle 4x^2 - 20x + 25 = 4x^2 - 10x - 10x + 25$

    $\displaystyle = (4x^2 - 10x) + (-10x + 25)$

    and do your factoring from there.

    -Dan
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