let 'x' and 'y' be the numbers

sum of two number is 18

x + y = 18

y = 18 - x

sum of their square is 290

x^2 + y^2 = 290

plug y = 18 - x into the last equation

x^2 + (18 - x)^2 = 290

by solving we find

x1 = 17

x2 = 1

click here to see the step by step solution of the equation

the two numbers are 17 and 1

indeed 1+ 17 = 18

17^2 + 1^2 = 289 + 1 = 290

the numbers are 17 and 1.