Define Equations

Let the two numbers be $x$ and $y$. We are given that their sum is $50$ and difference is $10$.

$x + y = 50, \quad x - y = 10$

Apply Algebraic Identity

We can find the product $xy$ using the identity for the difference of squares or standard expansion of $(x+y)^2$ and $(x-y)^2$.

$(x+y)^2 - (x-y)^2 = 4xy$

Substitute Values

Substitute the given values of sum ($50$) and difference ($10$) into the equation.

$50^2 - 10^2 = 4xy$

Calculate the Product

Simplify the squares and solve for $xy$.

$2500 - 100 = 4xy \implies 2400 = 4xy \implies xy = 600$