[Surds] What am I doing wrong?? :( 10 points

I know that: If A + √B = C + √D then A = C, B = D

The question is: Solve for x and y
√2 all over 2√2 - 2√3 = x + y√6.

I keep on getting x = -1, y = 1/2
but when you put it in the calculator, -1 + 1/2√6 does not = √2 / 2√2 - 2√3.
I rationalize it first and then I equate it with the above formula. So what am I doing wrong please??

sqrt(2) / (2 * sqrt(2) - 2 * sqrt(3))

rationalize the denominator:

sqrt(2) * (2 * sqrt(2) + 2 * sqrt(3)) / ((2 * sqrt(2) - 2 * sqrt(3)) * (2 * sqrt(2) + 2 * sqrt(3))) =>
(2 * 2 + 2 * sqrt(2 * 3)) / (4 * 2 - 4 * 3) =>
(4 + 2 * sqrt(6)) / (-4) =>
(-1) - (1/2) * sqrt(6)

y = -1/2, not +1/2. It was a simple mistake (and those are the hardest ones to see)