All are non-negative. The sum of roots is $ 0 + 4 + 9 = 13 $.

["Title: Understanding Non-Negative Roots: Why Their Sum Equals 13", "When exploring mathematical equations, especially quadratic ones, one intriguing concept is the sum of roots — particularly in contexts where all roots are non-negative. In this article, we’ll explore the idea that all roots are non-negative and demonstrate how their sum, specifically $ 0 + 4 + 9 = 13 $, reflects meaningful properties in algebra and real-world applications.", "### What Does “All Roots Are Non-Negative” Mean?", "In algebra, the roots (or solutions) of a quadratic equation $ ax^2 + bx + c = 0 $ can be real and complex, positive, negative, or zero. Saying all roots are non-negative means that every solution $ x \geq 0 $. This condition is not only mathematically interesting but also crucial in fields like engineering, physics, and economics, where negative values may represent non-physical quantities.", "### The Sum of Roots Formula", "For a quadratic equation with two real roots $ r_1 $ and $ r_2 $, Vieta’s formulas tell us that:", "> Sum of roots: $ r_1 + r_2 = -\frac{b}{a} $", "This elegant relationship connects coefficients to solutions directly. When all roots are non-negative, this sum is a non-negative number — often zero or positive.", "### The Case: Roots 0, 4, and 9", "Let’s consider the specific example where the roots are $ r_1 = 0 $, $ r_2 = 4 $, and $ r_3 = 9 $ (note: for a quadratic equation, only two roots exist, but this example extends the idea using three non-negative values summing to 13).", "Sum:\n[\n0 + 4 + 9 = 13\n]", "Even though this total arises from more than two non-negative numbers (interpreted here as contributing to a cumulative measure), it illustrates how non-negative values combine additively. In some applied models—such as total energy contributions, population growth rates, or cumulative scalings—adding non-negative components naturally leads to non-negative totals.", "### Why This Matters", "Knowing the sum of non-negative roots helps in:", "- Model Validation: Ensuring solutions reflect physical realities (e.g., no negative measurements).\n- Optimization Problems: Simplifying constraints in economic or engineering systems where only positive outcomes are meaningful.\n- Data Interpretation: Understanding aggregate behaviors where each component is inherently non-negative.", "### Conclusion", "While a quadratic has only two roots, the idea of sum of non-negative roots — such as $ 0 + 4 + 9 = 13 $ — reveals deeper insights into mathematical modeling. These principles empower accurate analysis across sciences and applied fields, confirming that non-negative roots lead to meaningful, real-world summed outcomes.", "---", "Key Takeaways:\n- All roots non-negative means solutions are ≥ 0.\n- Vieta’s formula gives root sum as $ -b/a $.\n- Non-negative roots sum to a meaningful total — e.g., 13 in this example.\n- Applicable in physics, finance, and engineering contexts.", "Explore more about quadratic roots and non-negative constraints to enhance your mathematical fluency and problem-solving skills.", "---", "Keywords:\nsum of roots, non-negative roots, quadratic equation, Vieta’s formulas, real roots, mathematical modeling, algebra education, quadratic sum = 13, non-negative root constraints."]