From Vieta's formulas, the sum of the roots \( -\frac{b}{a} = 3 + (-2) = 1 \), so \( b = -1 \).

["From Vieta’s Formulas: Deriving the Sum and Product of Roots Easily", "When working with quadratic equations, understanding the relationship between the coefficients and the roots is essential for solving and analyzing polynomial behavior. One of the most powerful tools in this regard comes from Vieta’s formulas, which connect the coefficients of a polynomial to sums and products of its roots—without needing to explicitly solve for the roots.", "### Understanding Vieta’s First Formula for Quadratic Roots", "For a standard quadratic equation:\n[\nax^2 + bx + c = 0\n]\nVieta’s formulas tell us:\n- The sum of the roots, denoted ( r_1 + r_2 ), equals ( -\frac{b}{a} ).\n- The product of the roots, ( r_1 \cdot r_2 ), equals ( \frac{c}{a} ).", "In particular, the sum formula is:\n[\nr_1 + r_2 = -\frac{b}{a}\n]", "This relationship lets us instantly compute the sum of roots from the equation’s coefficients alone.", "### Applying the Concept: Sum of Roots Equals 3 + (−2) = 1", "Suppose we are given a quadratic equation and told that the sum of its roots is ( 3 + (-2) = 1 ). Using Vieta’s sum formula:\n[\n-\frac{b}{a} = 1\n]\nThis means:\n[\n\frac{b}{a} = -1\n]", "From here, since we assume ( a = 1 ) (common in monic quadratics unless otherwise specified), simplifying gives:\n[\nb = -1\n]", "Thus, knowing that the sum of roots is ( 1 ), Vieta’s method efficiently leads us to conclude ( b = -1 ), linking coefficients and roots with just algebraic manipulation.", "### Why This Matters — Practical Implications", "Using Vieta’s formulas saves time and reduces errors when analyzing quadratic equations:\n- You avoid solving quadratic equations explicitly.\n- You quickly validate solutions or infer properties like symmetry or location of roots.\n- For higher-degree polynomials, the logic extends naturally, where sum of roots becomes the negative coefficient of the second-highest degree divided by the leading coefficient.", "### Conclusion", "Vieta’s formulas provide an elegant framework to explore the deep connection between a polynomial’s coefficients and its roots. In our example—using ( -\frac{b}{a} = 3 + (-2) = 1 )—we leveraged this relationship to deduce that ( b = -1 ), showcasing how powerful algebra can be for root-based insights with minimal computation.", "Whether designing problems, teaching algebra, or solving real-world models, applying Vieta’s formulas makes it easier to “read” equations and understand the invisible patterns within.", "---", "Key Takeaways:\n- Vieta’s sum formula for quadratic roots: ( r_1 + r_2 = -\frac{b}{a} )\n- From ( r_1 + r_2 = 3 + (-2) = 1 ), we find ( b = -1 )\n- Use Vieta’s to unlock root relationships without solving equations", "---", "Keywords for SEO:\nVieta’s formulas, sum of roots, quadratic equations, derive sum of roots, math tutorial, algebraic formulas, roots and coefficients, polynomial relationships."]