Results 1 to 6 of 6
Like Tree2Thanks
  • 1 Post By Hartlw
  • 1 Post By Hartlw

Math Help - find all possible values

  1. #1
    Member
    Joined
    Nov 2006
    Posts
    152

    find all possible values

    Let  a,b,c,d positive real numbers such that

     a+b+c+d=12 , and

    abcd =27+ab+ac+ad+bc+bd+cd

    find all possible values of

    a,b,c,d
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Jun 2013
    From
    Bangalore
    Posts
    63
    Thanks
    1

    Re: find all possible values

    Solution:
    Given a+b+c+d=12
    and abcd=27+ab+ac+ad+bc+bd+cd
    then a,b,c and d =?
    let a=b=c=d=3
    then 3+3+3+3=12
    LHS => abcd=3*3*3*3=81
    RHS=> 27+3*3+3*3+3*3+3*3+3*3+3*3=81
    hence a=b=c=d=12
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Re: find all possible values

    Solve second eq for a, substitute into first eq, and multiply through by bcd to get:
    (…) = 12bcd -27.
    The left side has to be positive so eqs have a sol for all b,c,d s.t. 12bcd>27
    Last edited by Hartlw; September 18th 2013 at 07:03 AM.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,929
    Thanks
    333
    Awards
    1

    Re: find all possible values

    Quote Originally Posted by brosnan123 View Post
    Solution:
    Given a+b+c+d=12
    and abcd=27+ab+ac+ad+bc+bd+cd
    then a,b,c and d =?
    let a=b=c=d=3
    then 3+3+3+3=12
    LHS => abcd=3*3*3*3=81
    RHS=> 27+3*3+3*3+3*3+3*3+3*3+3*3=81
    hence a=b=c=d=12
    Typo in line 8:
    a = b = c = d = 3

    But how to show that there aren't any other solutions? This might not be all of them.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Re: find all possible values

    Quote Originally Posted by Hartlw View Post
    Solve second eq for a, substitute into first eq, and multiply through by bcd to get:
    (…) = 12bcd -27.
    The left side has to be positive so eqs have a sol for all b,c,d s.t. 12bcd>27
    Above is correct. However, after placing one condition in another to get a condition, you still have to retain one of the original two conditions. Sorry.
    Using a+b+c+d=12, a = 12-(b+c+d), a>0

    So all a,b,c,d st:
    12bcd>27
    b+c+d<12
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Aug 2010
    Posts
    961
    Thanks
    98

    Re: find all possible values

    Quote Originally Posted by Hartlw View Post
    Above is correct. However, after placing one condition in another to get a condition, you still have to retain one of the original two conditions. Sorry.
    Using a+b+c+d=12, a = 12-(b+c+d), a>0

    So all a,b,c,d st:
    12bcd>27
    b+c+d<12
    The above is correct, but it doesn’t work, I tried it- it is necessary but not sufficient. A sufficient conditon is given below following an example.

    a+b+c+d=12
    abcd=27+a(b+c+d)+b(c+d)+cd.

    Example:
    Assume a=4, and b=2. Then
    1) c+d=6
    8cd=27+4(8)+cd
    2) 7cd=59

    1) and 2) give a quadratic equation with solution c=.756 and d=5.244

    In general:
    Assume a and b given. Then:
    3) c+d=12-(a+b)
    (ab-1)cd=27+a(b+(12-(a+b))+b[12-(a+b)], or:
    4) cd=k

    3) and 4) give a quadratic eq for c which has a solution if (c+d)^2>4k (not very nice).

    Topsquark: I pissed in your ear and told you it was raining. My sincerest apologies. Feel free to remove the thanks- I appreciate the gesture.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: January 28th 2010, 01:39 AM
  2. find values
    Posted in the Algebra Forum
    Replies: 5
    Last Post: September 19th 2009, 08:33 AM
  3. Find the values of F1 and F2
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: August 19th 2009, 10:20 AM
  4. find values of p:
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: June 22nd 2009, 10:09 AM
  5. Replies: 1
    Last Post: May 24th 2009, 05:16 AM

Search Tags


/mathhelpforum @mathhelpforum