Solution: Let the populations be $ a $ and $ b $. Given $ a + b = 10 $ and $ a^2 + b^2 = 50 $. First, compute $ ab $ using $ (a + b)^2 = a^2 + 2ab + b^2 $. Substituting, $ 100 = 50 + 2ab $, so $ ab = 25 $. The sum of cubes is $ a^3 + b^3 = (a + b)^3 - 3ab(a + b) = 1000 - 3 \cdot 25 \cdot 10 = 1000 - 750 = oxed{250} $.

["Optimized Solution to a Quadratic Population Problem Using Algebraic Identities", "In many mathematical modeling scenarios, especially in data science and operations research, problems involving sums and sums of squares of populations arise naturally. Consider a scenario where two populations have sizes $ a $ and $ b $, with known total $ a + b = 10 $ and sum of squares $ a^2 + b^2 = 50 $. Finding key metrics such as $ ab $ (the product of population sizes) and $ a^3 + b^3 $ (useful in modeling growth dynamics) requires smart algebraic manipulation—not brute force.", "Let’s walk through the elegant solution using a fundamental identity:", "$$\n(a + b)^2 = a^2 + 2ab + b^2\n$$", "We are given:\n- $ a + b = 10 $ → $ (a + b)^2 = 100 $\n- $ a^2 + b^2 = 50 $", "Substitute into the identity:", "$$\n100 = 50 + 2ab\n$$", "Solving for $ ab $:", "$$\n2ab = 100 - 50 = 50 \quad \Rightarrow \quad ab = 25\n$$", "This product $ ab = 25 $ reveals critical information—such as the geometric relationship between the populations under fixed sum and sum of squares.", "Next, we compute $ a^3 + b^3 $, which often serves in cubic growth models. Using the identity:", "$$\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n$$", "Substitute known values:\n- $ a + b = 10 $\n- $ ab = 25 $", "$$\na^3 + b^3 = 10^3 - 3 \cdot 25 \cdot 10 = 1000 - 750 = \boxed{250}\n$$", "This computed value allows for predictive modeling in scenarios such as total resource allocation scaled by cubic demand functions, or combined performance metrics across intersecting groups.", "Conclusion:\nBy leveraging the identity $ (a + b)^2 = a^2 + 2ab + b^2 $, we efficiently determined $ ab = 25 $ without solving a quadratic equation. Using this, we then calculated $ a^3 + b^3 = 250 $—a key result for advanced mathematical and data-driven applications. This approach highlights the power of algebraic identities in simplifying complex problems involving two-variable populations.", "Keywords:\nPopulation algebra, $ a + b = 10 $, $ a^2 + b^2 = 50 $, compute $ ab $, sum of cubes, $ a^3 + b^3 = 250 $, algebraic identities, quadratic populations, mathematical modeling."]