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?
Let the two integers be $x$ and $y$. We are given that $x^2 + y^2 = 193$ and that $xy = 84$. We want to find $x + y$. Note that $(x + y)^2 = x^2 + y^2 + 2xy = 193 + 2\cdot 84 = 361$. Taking the square root of 361, we see that $x + y = \boxed{19}$.
19
Level 2
Algebra
Problem: 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?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the sum of the two integers, let the integers be denoted as x and y. We know from the problem that:

1. x^2 + y^2 = 193 (the sum of their squares)
2. xy = 84 (the product of the integers)

We can use the identity that relates the sum of squares to the square of the sum:

x^2 + y^2 = (x + y)^2 - 2xy

Substituting the known values into this identity gives:

193 = (x + y)^2 - 2(84)

Calculating the product term:

2(84) = 168

Now substituting this back into the equation:

193 = (x + y)^2 - 168

Rearranging this equation to isolate (x + y)^2:

(x + y)^2 = 193 + 168
(x + y)^2 = 361

Taking the square root of both sides yields:

x + y = √361
x + y = 19

Thus, the sum of the two integers is 19.<|END_OF_LATENT|> The final answer is $19$.