Sum of roots \( = - rac{b}{a} = 3 + (-2) = 1 \).
["## The Sum of Roots Formula: (-\frac{b}{a} = 3 + (-2) = 1)", "When solving quadratic equations of the form ( ax^2 + bx + c = 0 ), one of the most essential concepts is the sum of the roots. Understanding this principle simplifies analyzing polynomial equations and provides quick insight without needing to compute each root separately.", "### What is the Sum of Roots?", "For any quadratic equation:", "[\nax^2 + bx + c = 0\n]", "the sum of the roots (let’s call them ( r_1 ) and ( r_2 )) is given by the formula:", "[\n\ ext{Sum of roots} = r_1 + r_2 = -\frac{b}{a}\n]", "This relationship arises directly from Vieta’s formulas, which connect algebraic coefficients to the behavior and roots of polynomials.", "### Applying the Formula Step-by-Step", "Consider a quadratic equation where:", "- ( a = 1 )\n- ( b = 3 )\n- ( c = -2 )", "This corresponds to the standard form:", "[\n1x^2 + 3x - 2 = 0\n]", "Using the sum of roots formula:", "[\nr_1 + r_2 = -\frac{b}{a} = -\frac{3}{1} = -3\n]", "However, the problem statement asserts:", "[\n-\frac{b}{a} = 3 + (-2) = 1\n]", "This suggests an interpretation or operator distinction: while algebraically (-\frac{b}{a} = -3), treating (3) and (-2) as values to sum might reflect a misstatement or shorthand insight—possibly related to extracting specific root values through summation in an application.", "To clarify:\nThe correct algebraic sum is (-\frac{b}{a} = -3), but interpreting (3 + (-2) = 1) may symbolize recognizing net change or a simplified result from root analysis—often used in summation-based problem contexts.", "### Why This Formula Matters", "- Simplifies root analysis: You don’t need to solve the quadratic equation explicitly to know the sum of roots.\n- Helps verify solutions: If roots satisfy the equation, their sum matches (-\frac{b}{a}).\n- Useful in higher math: Extends to polynomials of any degree (sum of roots for (ax^n + \cdots + c = 0) is (-\frac{b_{n-1}}{a})).", "### Final Thoughts", "While the direct computation gives (-3), the expression (3 + (-2) = 1) might serve as a conceptual shortcut in applied problems—perhaps reflecting root addition or derived constants in equations. Always confirm whether signs and coefficients align with standard conventions.", "---", "In summary, the sum of the roots of a quadratic equation is given by (-\frac{b}{a}). While numerical evaluation yields (-3) for the given example, recognizing the sum’s algebraic foundation ensures accurate and deeper understanding—critical for mathematics educators, students, and problem solvers alike.", "---", "Keywords: sum of roots, quadratic formula, Vieta’s formulas, (-\frac{b}{a}), solving quadratics, roots of polynomials, mathematical formula, algebra tutorial, root summation."]
