Shouldn't the answer also include 9? Since the sum of their roots (36 + 9) equals 9?

You mean a triangle whose base is 6 and height is 3? The hypotenuse is \sqrt{45} which is not what we’re looking for.
(you can’t do this the other way round: recall that it’s a^2 + b^2 = c^2, not \sqrt{a} + \sqrt{b} = \sqrt{c})