SOLUTION: what is the smallest positive integer of k such that the equation sqrt(x - 127) + sqrt(k - x) = 13 has at least one real solution for x?

Isolate a square root term:



Square both sides:







Add 127 to both sides



Add x to both sides:



Isolate the square root term



Square both sides:









Change all signs







This quadratic will have a real solution if and only if
its discriminant is not negative:

Discriminant = B²-4AC =











This discriminant will be nonnegative if







So the answer is the next integer after 211.5 which is 212.

Edwin