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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

If x and y are rational numbers satisfying:

what is the value of x+y?

Square both sides and simplify:

Since x and y are rational numbers the irrational quantity on the left side must equal the irrational quantity on the right side and similarly for the rational quantities. This results in two equations: one for the rational parts and one for the irrational parts.

m4

Solve the system of two equations:

by solving for y in the first equation and substituting in the second equation:

and obtain a bi-quadratic equation:

We can solve this equation similarly to a quadratic equation but not forgetting that it will usually have four roots.

Since x has been specified as rational, only the rational solutions qualify.

Since x and y are either both positive or both negative, the possible sums for x and y are in the set: