### Instruction: The sum of the squares of two positive integers is 193. The product of the two integers is 84. What is the sum of the two integers?### Response: This problem involves two positive integers, let's call them x and y, and their squares and product.
I want to find their sum, x + y.
I notice that the sum of the squares, x^2 + y^2, is given as 193, and the product, xy, is given as 84.
This makes me think of the algebraic identity (x + y)^2 = x^2 + 2xy + y^2.
If I apply this identity to the given information, I get that (x + y)^2 = 193 + 2(84) = 361.
Taking the square root of both sides, I get that x + y = sqrt(361) = 19.
This is the answer I was looking for.