let 'x' and 'y' be the two positive numbers

the difference of two positive numbers is 8

x - y = 8

x = 8 + y

the sum of their squares is 130

x^2 + y^2 = 130

we plug x = 8 + y into the last equation and get

(8 + y)^2 + y^2 = 130

by solving the quadratic equation we find and consider the positive roots

y = 3

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x = 3 + 8

x = 11

the numbers are 11 and 3.