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Real Resolution (Posted on 2014-01-16) Difficulty: 3 of 5
Determine all possible real solutions to this system of equations:

A+√(B*C) = 29, B+√(C*A) = 31, C+√(A*B) = 37

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
Rating: 3.5000 (2 votes)

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Solution More solutions Comment 2 of 2 |
An integer solution has already been posted, but I think I’ve found the other
solutions and would welcome any news of a shorter method.

It’s clear that A, B, C must have the same sign for real square roots, so

substituting B = k2A is permissible and the three equations then become:

A + k*sqrt(AC) = 29;     k2A + sqrt(AC) = 31;      C + kA = 37

Eliminating sqrt(AC) between the first two gives A = (31k – 29)/(k3 – 1),

thus B = k2(31k – 29)/(k3 – 1) and, from the third equation, after some

rearrangement:  C = (37k3 - 31k2 +29k – 37)/(k3 – 1).

Substituting these results into the original equation, B + sqrt(CA) = 31,

after much algebra gives:  153k4 - 1017k3 +1798k2 – 994k + 56 = 0

which factorises to (3k – 4)(51k3 -271k2 + 238k – 14) = 0.

The linear factor yields k = 4/3 which gives the integer solution already
posted, while the cubic factor gives three irrational roots, two of which
give values of A, B and C that satisfy the original problem, as follows:

                                    A                      B                      C

k = 4/3                         9                      16                     25

k = 0.0633369..             27.0434..           0.108486..         35.2872..

k = 4.22442..                 1.37061..           24.4596..           31.2099..

k = 1.02597..                  not a valid solution.



  Posted by Harry on 2014-01-31 17:17:59
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