perplexus.info :: Just Math : Real and Cyclic Sum

Real and Cyclic Sum (Posted on 2016-08-10)

Determine all possible real solutions to this system of equations:

A+B+C+D = 5, and:
A*B+B*C+C*D+D*A = 4, and:
A*B*C+B*C*D+C*D*A+D*A*B= 3, and:
A*B*C*D= -1

Prove that there are no others.

See The Solution Submitted by K Sengupta

Rating: 4.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

adding on

| Comment 2 of 3 |

Good work, Harry.

You can extend your method to the third equation, then substitute the values you found for (A+C) and (B+D).

AC(B+D)+BD(A+C)=3 becomes 4(AC) + BD=3 using your first values. Then you can solve that simultaneously with equation 4 to get values for AC and BD.

At that point there's enough information to evaluate A and C.

I don't have the time now for a full listing, but I've found these other values for A:

(1+sqrt(5))/2

(1-sqrt(5))/2

2+sqrt(5)

2-sqrt(5)

(4+sqrt(17))/2

(4-sqrt(17))/2

Posted by xdog on 2016-08-11 18:20:34

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