Answer
AD = 10; BD = 5
Work Step by Step
Using the book's recommendation to use AD = 2x and BD = x as well as the Pythagorean theorem, we have:
$(2x)^2 - 6^2 = (x+3)^2 \\ 3x^2 - 6x - 45 = 0 \\ 3(x-5)(x+3) = 0 \\ x=5$
Thus:
$AD =2(5) = 10 \\ BD = 5 $