A quadratic equation \( ax^2 + bx + c = 0 \) has roots \( \alpha \) and \( \beta \). If \( a = 2 \), \( b = -7 \), and \( c = 3 \), find the sum and product of the roots.
["Understanding the Sum and Product of Roots in a Quadratic Equation", "When studying quadratic equations of the form ( ax^2 + bx + c = 0 ), one of the most valuable concepts is understanding the roots and their properties. For any quadratic equation with real coefficients, the roots—denoted as ( \alpha ) and ( \beta )—follow well-defined relationships involving the coefficients. When the leading coefficient ( a = 2 ), ( b = -7 ), and ( c = 3 ), these relationships simplify, allowing us to quickly compute the important sum and product of the roots.", "### The Standard Quadratic Formula and Root Relationships", "The roots of the equation ( ax^2 + bx + c = 0 ) are given by the quadratic formula:\n[\n\alpha = \frac{-b + \sqrt{b^2 - 4ac}}{2a}, \quad \beta = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\n]\nHowever, rather than computing the roots explicitly, we can use elegant formulas derived from Vieta’s relations, which directly link the coefficients of the equation to the sum and product of the roots.", "### Sum and Product of Roots Formula", "For any quadratic equation ( ax^2 + bx + c = 0 ), the sum and product of the roots are:\n- Sum: ( \alpha + \beta = -\dfrac{b}{a} )\n- Product: ( \alpha \beta = \dfrac{c}{a} )", "These formulas hold regardless of whether the roots are real or complex, provided ( a <br/>\neq 0 ).", "### Plugging in the Given Values", "We are given:\n- ( a = 2 )\n- ( b = -7 )\n- ( c = 3 )", "Compute the sum of the roots:\n[\n\alpha + \beta = -\frac{b}{a} = -\left( \frac{-7}{2} \right) = \frac{7}{2}\n]", "Compute the product of the roots:\n[\n\alpha \beta = \frac{c}{a} = \frac{3}{2}\n]", "### Conclusion", "Therefore, for the quadratic equation ( 2x^2 - 7x + 3 = 0 ), the sum of the roots is ( \frac{7}{2} ), and the product is ( \frac{3}{2} ). This method avoids lengthy calculations and directly leverages the powerful relationships between coefficients and roots in quadratic equations.", "Understanding these properties not only simplifies problem-solving but also strengthens foundational mathematical insight—especially useful in algebra, calculus, and beyond. If you're working with any quadratic equation, remember: the sum of the roots is ( -\frac{b}{a} ), and the product is ( \frac{c}{a} ).", "Keywords: quadratic equation, roots, sum of roots, product of roots, Vieta's formulas, equation ax² + bx + c = 0, a = 2, b = -7, c = 3, -b/a, c/a, algebra, mathematics, solving quadratics."]
