The sum of the roots is equal to the negative of the coefficient of \( x \), which is \(-b\), and the product of the roots is \( c \). If the roots are in the ratio 3:2, find the values of \( b \) and \( c \).

["The Fundamental Vieta’s Formulas: How Root Ratios Reveal Coefficients (b) and (c)", "Understanding the relationship between the coefficients of a polynomial and the roots is a cornerstone of algebra. One of the most powerful results is Vieta’s formulas, which connect the roots of a polynomial directly to its coefficients. For a quadratic equation, these relationships become particularly intuitive and widely used. This article explores how the sum and product of roots relate to coefficients, focusing on the key insight: if the roots are in a 3:2 ratio, we can determine exact values for (b) and (c).", "---", "### Vieta’s Formulas for a Quadratic Equation", "Consider a general quadratic polynomial:", "[\nax^2 + bx + c = 0\n]", "Let the roots be (r_1) and (r_2). Vieta’s formulas tell us:", "- Sum of roots: ( r_1 + r_2 = -\frac{b}{a} )\n- Product of roots: ( r_1 \cdot r_2 = \frac{c}{a} )", "When the leading coefficient (a = 1) (the standard form (x^2 + bx + c = 0)), these simplify beautifully to:", "[\nr_1 + r_2 = -b \quad \ ext{and} \quad r_1 \cdot r_2 = c\n]", "This simplification allows us to directly link the sum and product of roots to the coefficients (b) and (c). This relationship becomes especially useful when the roots are known to be in a particular ratio — such as 3:2.", "---", "### Given: Roots in a 3:2 Ratio", "Suppose the two roots are in the ratio:", "[\nr_1 : r_2 = 3 : 2\n]", "To simplify calculations, let:", "[\nr_1 = 3k \quad \ ext{and} \quad r_2 = 2k \quad \ ext{for some positive constant } k\n]", "Now compute the sum and product of the roots:", "- Sum:\n[\nr_1 + r_2 = 3k + 2k = 5k \Rightarrow -b = 5k \Rightarrow b = -5k\n]", "- Product:\n[\nr_1 \cdot r_2 = (3k)(2k) = 6k^2 \Rightarrow c = 6k^2\n]", "So far, we have expressions for (b) and (c) in terms of (k):", "[\nb = -5k, \quad c = 6k^2\n]", "But to find specific numbers, we need one additional constraint — otherwise, (k) remains a free parameter. This is where the polynomial’s leading coefficient matters.", "---", "### Incorporating the Leading Coefficient", "Recall that Vieta’s formulas assume a specific leading coefficient. If the polynomial is monic ((a = 1)), then our earlier expressions are exact. However, if the polynomial has a general leading coefficient (a), the relationships scale:", "[\nr_1 + r_2 = -\frac{b}{a} \Rightarrow 5k = -\frac{b}{a} \Rightarrow b = -5ak\n]\n[\nr_1 r_2 = \frac{c}{a} \Rightarrow 6k^2 = \frac{c}{a} \Rightarrow c = 6ak^2\n]", "Here, (k) is still arbitrary unless tied to another condition — like the actual polynomial or its value at a point. But in standard problems, especially in OP (Opportunity Points) related to root analysis, assuming (a = 1) is common unless stated otherwise.", "Thus, assuming (a = 1):", "[\nb = -5k, \quad c = 6k^2\n]", "Without loss of generality, choose (k = 1) to find a specific solution (since ratios are invariant under scaling, but (b) and (c) depend on scale).", "---", "### Final Values (with (k = 1))", "[\nb = -5(1) = -5, \quad c = 6(1)^2 = 6\n]", "So, for a quadratic polynomial with roots in a 3:2 ratio and monic leading coefficient:", "- ( b = -5 )\n- ( c = 6 )", "The corresponding quadratic equation is:", "[\nx^2 - 5x + 6 = 0\n]", "Factoring:", "[\n(x - 3)(x - 2) = 0 \Rightarrow \ ext{Roots: } 3, 2 \Rightarrow \ ext{Ratio } 3:2, \ ext{ Sum } 5 = -(-5), \ ext{ Product } 6 = 6\n]", "All match!", "---", "### General Insight", "For any quadratic with roots in ratio 3:2:", "[\nb = -5k, \quad c = 6k^2\n]", "But without fixing (k) (typically (a = 1)), (b) and (c) scale together. To pinpoint exact numbers, more context is needed — such as the value of the polynomial at (x = 1), or a specific leading coefficient.", "However, the elegance lies in how ratios alone reveal structure: knowing the root relationship lets us express (b) and (c) entirely in terms of a single variable, unlocking deeper algebraic insights.", "---", "### Why This Matters", "Vieta’s formulas turn abstract root properties into concrete coefficient equations. Whether solving quadratics, analyzing stability in systems, or modeling real-world data, knowing how roots “encode” the coefficients enables powerful calculations and predictions.", "In this case, the 3:2 root ratio led directly to:", "[\n\boxed{b = -5,\quad c = 6}\n]", "(assuming monic polynomial and (k = 1)) — a clean, elegant solution rooted in fundamental algebra.", "---", "Keywords: Vieta’s formulas, sum of roots = (-b), product of roots = (c), roots in ratio 3:2, quadratic equation, algebraic relationships, solving polynomials, monic polynomial, k-value, root-to-coefficient conversion.", "Meta Description:\nDiscover how Vieta’s formulas connect root ratios to coefficients—especially when roots are in 3:2 ratio. Learn how (b = -5k) and (c = 6k^2), and find exact values when (a = 1) and (k = 1). Perfect for algebra students and problem solvers."]