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Math Help - A Generalized Problem Problem

  1. #1
    Newbie
    Joined
    May 2011
    Posts
    15

    A Generalized Problem Problem

    Hey all,

    So in the shower, I found myself trying to generalize a problem. I was discussing with my students earlier how the two numbers, x and y, that both add and multiply to 4
    are both 2. In short, 2+2 and 2*2 are both 4. I was trying to generalize this pattern to work for numbers other than 4. Here is the system I am starting with.

    x + y = A
    xy = A

    Step 1)
    x = A-y, so (A-y)y = A, so Ay - y^2 = A, so -y^2 + Ay - A = 0 (quadratic formula time..)

    a= -1
    b = A
    c = -A

    {-A + sqrt[A^2 -4(-1)(-A)] } / [2(-1)]

    Simplified..

    {-A + sqrt(A^2 - 4A)} / -2

    Simplified..

    A/2 + sqrt(A^2 - 4A)

    To check, I plugged in A = 4 (because then I know my answer should be 2)

    4/2 + sqrt(16-16) = 2

    To see what the problem would be with A = 6..

    6/2 + sqrt(36 - 24) = 3 + sqrt(12)

    So, I plugged this into my original system to solve for the other variable.

    [3 + sqrt(12)] + y = 6, so y must be 6 - 3 - sqrt(12) = 3 - sqrt(12).

    So, x + y = 3+sqrt(12) + 3 - sqrt (12), which is 6. First equation checks.

    Second equation is what is giving me a headache...

    xy = 6

    (3 + sqrt(12))*((3 - sqrt(12)) = 6

    9 - 3sqrt(12) + 3sqrt(12) - 12 = 6

    9 - 12 = 6 (does not work)

    Which error am I making?

    Cheers,
    Ted
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  2. #2
    Member HappyJoe's Avatar
    Joined
    Sep 2010
    From
    Denmark
    Posts
    234
    Hello.

    I have made a comment inside of your work:

    Quote Originally Posted by rdtedm View Post
    Hey all,

    So in the shower, I found myself trying to generalize a problem. I was discussing with my students earlier how the two numbers, x and y, that both add and multiply to 4
    are both 2. In short, 2+2 and 2*2 are both 4. I was trying to generalize this pattern to work for numbers other than 4. Here is the system I am starting with.

    x + y = A
    xy = A

    Step 1)
    x = A-y, so (A-y)y = A, so Ay - y^2 = A, so -y^2 + Ay - A = 0 (quadratic formula time..)

    a= -1
    b = A
    c = -A

    {-A + sqrt[A^2 -4(-1)(-A)] } / [2(-1)]

    Simplified..

    {-A + sqrt(A^2 - 4A)} / -2

    Simplified..

    A/2 + sqrt(A^2 - 4A) It looks like you forgot to divide the 2 into the term with the square root, so it should have been A/2 + sqrt(A^2-4A)/(-2).

    To check, I plugged in A = 4 (because then I know my answer should be 2)

    4/2 + sqrt(16-16) = 2

    To see what the problem would be with A = 6..

    6/2 + sqrt(36 - 24) = 3 + sqrt(12)

    So, I plugged this into my original system to solve for the other variable.

    [3 + sqrt(12)] + y = 6, so y must be 6 - 3 - sqrt(12) = 3 - sqrt(12).

    So, x + y = 3+sqrt(12) + 3 - sqrt (12), which is 6. First equation checks.

    Second equation is what is giving me a headache...

    xy = 6

    (3 + sqrt(12))*((3 - sqrt(12)) = 6

    9 - 3sqrt(12) + 3sqrt(12) - 12 = 6

    9 - 12 = 6 (does not work)

    Which error am I making?

    Cheers,
    Ted
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2011
    Posts
    15
    I realized my mistake on my way to lunch. Thank you!

    Ted
    Follow Math Help Forum on Facebook and Google+

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