The difference between two positive integers is $6$ and their product is $40$. What is the sum of the integers?

Answer 1

Let the two positive integers be $x$ and $y$. We are given that $x-y=6$ and $xy=40$. Substituting $y = x-6$ into the second equation, you get $x(x-6) = 40$, which simplifies to $x^2 - 6x - 40 = 0$. This quadratic factors as $(x-10)(x+4)= 0$, so the possibilities for $x$ are 10 and -4. Since $x$ must be positive, we have $x = 10$, and so $y = x-6 = 4$. Thus, $x+y = \boxed{14}$.