What number should be added to both sides of the equation to complete the square x^2+3x=6

Take half of the coefficient of x:  It is 3, and half that is 3/2.

Then   x^2+3x=6  becomes:

           x^2+3x + (3/2)^2 =6 + (3/2)^2, and

             (x+3/2)^2 = 6 + 9/4

You were not asked to solve the equation, but why not do it for the practice?

Solve (x+3/2)^2 = 6 + 9/4 for x.  There will be 2 values.

The first one, i believe, would be correct. all the others have one thing in common, and that's that they have repeating numbers. in the second one, x and y have repeating 6's and 3's, and in the third, x and y have repeating 6's as well. in the last one, there are repeating 3's. the first one has no repeating numbers.