Let$\displaystyle a,b,c,d$ positive real numbers such that

$\displaystyle a+b+c+d=12$ , and

$\displaystyle abcd =27+ab+ac+ad+bc+bd+cd $

find all possible values of

$\displaystyle a,b,c,d$

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- Jan 24th 2008, 09:12 AMperashfind all possible values
Let$\displaystyle a,b,c,d$ positive real numbers such that

$\displaystyle a+b+c+d=12$ , and

$\displaystyle abcd =27+ab+ac+ad+bc+bd+cd $

find all possible values of

$\displaystyle a,b,c,d$ - Sep 16th 2013, 10:45 PMbrosnan123Re: 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 - Sep 18th 2013, 06:56 AMHartlwRe: 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 - Sep 18th 2013, 12:57 PMtopsquarkRe: find all possible values
- Sep 18th 2013, 02:06 PMHartlwRe: find all possible values
- Sep 19th 2013, 04:52 AMHartlwRe: find all possible values
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.