A correct solution came from Bradley University student Robert Andry.  Other correct solutions were submitted by Bill Webb, Ingrid Lidó,  Juan Carlos Marivela,  A. Teitelman, Dan Dima, Ahron Kahn, Iñigo Picaza, Lou Cairoli.

Equate each of the two terms equal to 1 and square both sides to get 

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Expand and cancel equal terms to get which implies one of the following: (i) a = b = 0; (ii) c = d = 0; (iii) x = -b/a; (iv) x = -d/c.  The first two cases are of no interest.  In case (iii), substitute the value for x into  to get that ad - bc = a; you get the same by substituting into the second radical equation.  The case (iv) yields the symmetric bc - ad = c.  

Our young man was lucky enough to have picked a = 3, b = 7, c = 1, d = 2 -- which happens to satisfy condition (iv) -- and the common solution x = -d/c = -2.

Aaron Kahn verified numerically that allowing a, b, c, d to take values from the set {-5,-4,-3,-2,-1,1,2,3,4,5}, which are fairly typical values for problems found in beginning algebra problems, the probability of coincidence is 5.56%, or about one in twenty problems.    

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