Let roots be $ r_1, r_2 $, then $ \frac{A + 2d}{A} = 1 + 2\frac{d}{A} = 1 + 2\frac{1}{x} $. Since product is $ x^2 = 28/9 $, $ x = \pm \sqrt{28}/3 $. Use symbolic. From $ 14x^2 + 42x + 27 = 0 $, divide by 14: $ x^2 + 3x + \frac{27}{14} = 0 $.

["Understanding Quadratic Roots and Exact Symbolic Solutions", "When solving quadratic equations of the form $ x^2 + bx + c = 0 $, especially when exploring root relationships and symbolic expressions, clarity in manipulation and substitution is essential. This article explores a key quadratic scenario involving extracted roots $ r_1 $ and $ r_2 $, leverages symbolic computation, and applies to a specific discriminant context.", "---", "### Let the roots be $ r_1 $ and $ r_2 $", "For a quadratic equation $ x^2 + bx + c = 0 $, the sum and product of roots are:", "[\nr_1 + r_2 = -b, \quad r_1 r_2 = c\n]", "In this setup, suppose $ b = 3 $ and $ c = \frac{27}{14} $, yielding the equation:", "[\nx^2 + 3x + \frac{27}{14} = 0\n]", "This aligns with our earlier form $ x^2 + 3x + \frac{27}{14} = 0 $. We proceed symbolically.", "---", "### Expressing the quadratic identity symbolically", "Given $ x^2 + 3x + \frac{27}{14} = 0 $, divide through by 14 to standardize:", "[\n\frac{14x^2 + 42x + 27}{14} = 0 \quad \implies \quad x^2 + 3x + \frac{27}{14} = 0\n]", "Now solve using the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here $ a = 1 $, $ b = 3 $, $ c = \frac{27}{14} $. Compute discriminant:", "[\n\Delta = 3^2 - 4 \cdot 1 \cdot \frac{27}{14} = 9 - \frac{108}{14} = 9 - \frac{54}{7} = \frac{63 - 54}{7} = \frac{9}{7}\n]", "Thus roots are:", "[\nx = \frac{-3 \pm \sqrt{\frac{9}{7}}}{2} = \frac{-3 \pm \frac{3}{\sqrt{7}}}{2}\n]", "Simplify:", "[\nx = -\frac{3}{2} \pm \frac{3}{2\sqrt{7}} = \frac{-3\sqrt{7} \pm \frac{3}{2}}{2\sqrt{7}} \quad \ ext{(rationalized form)}\n]", "More cleanly written via symbolic root expressions:", "[\nx = -\frac{3}{2} \pm \frac{3}{2\sqrt{7}}\n]", "These roots symbolize $ r_1 $ and $ r_2 $. By definition:", "[\nr_1 + r_2 = -3, \quad r_1 r_2 = \frac{27}{14}\n]", "---", "### Expressing $ \frac{A + 2d}{A} $ symbolically", "The expression appears in variants that relate $ A $ and $ d $, possibly representing coefficients or scaling factors embedded in root relationships. Assume $ A = 14 $ and $ d = x $ as defined earlier—reflecting the scaled discriminant information.", "Then:", "[\n\frac{A + 2d}{A} = 1 + 2\frac{d}{A} = 1 + 2\frac{x}{14} = 1 + \frac{x}{7}\n]", "But from earlier discriminant-derived form $ x = -\frac{3}{2} \pm \frac{3}{2\sqrt{7}} $, substitute:", "[\n\frac{x}{7} = \frac{-3}{14} \pm \frac{3}{14\sqrt{7}}\n]", "Thus:", "[\n\frac{A + 2d}{A} = 1 + \left( -\frac{3}{14} \pm \frac{3}{14\sqrt{7}} \right) = \frac{11}{14} \pm \frac{3}{14\sqrt{7}}\n]", "This symbolic form reflects a precise, rational-scaled transformation of the root-variable ratio, rooted in exact algebra.", "---", "### Final algebraic evaluation", "Given $ x = \pm \sqrt{\frac{28}{9}} = \pm \frac{2\sqrt{7}}{3} $ — wait, contradiction emerges: earlier discriminant gave $ \Delta = \frac{9}{7} $, not $ \frac{28}{9} $! Let’s reconcile.", "Rechecking: The problem states “Since product is $ x^2 = \frac{28}{9} $”, so suppose $ x^2 = \frac{28}{9} \Rightarrow x = \pm \frac{2\sqrt{7}}{3} $", "But from $ x^2 + 3x + \frac{27}{14} = 0 $, solved roots differ — thus contradiction suggests separate or related but shifted contexts.", "However, embracing the stated product condition: let $ x^2 = \frac{28}{9} $, so $ x = \pm \frac{2\sqrt{7}}{3} $. Then indeed $ x^2 = \frac{4 \cdot 7}{9} = \frac{28}{9} $, consistent.", "Now plug into symbolic identity:", "[\n\frac{A + 2d}{A} = 1 + 2\frac{d}{A}\n]", "With $ A = 14 $, $ d = x = \pm \frac{2\sqrt{7}}{3} $:", "[\n\frac{A + 2d}{A} = 1 + 2 \cdot \frac{\pm \frac{2\sqrt{7}}{3}}{14} = 1 \pm \frac{4\sqrt{7}}{3 \cdot 14} = 1 \pm \frac{2\sqrt{7}}{21}\n]", "So symbolic expressions become:", "[\n\boxed{1 \pm \frac{2\sqrt{7}}{21}}\n]", "---", "### Conclusion", "Through symbolic manipulation, we relate quadratic identity to root products via algebraic structure, even under transformed conditions. The expression $ \frac{A + 2d}{A} $, tied to discriminant scaling and root relations, reveals precise ratios through substitution and rationalization. When $ x^2 = \frac{28}{9} $ gives $ x = \pm \frac{2\sqrt{7}}{3} $, the identity yields:", "[\n1 \pm \frac{2\sqrt{7}}{21}\n]", "This demonstrates how symbolic root expressions connect naturally to rationalized, scaled coefficients—enhancing depth in algebraic modeling and verification.", "---", "Keywords: quadratic roots, symbolic algebra, discriminant calculation, root relationships, $ A + 2d $, $ \frac{A + 2d}{A} $, $ x = \pm \frac{2\sqrt{7}}{3} $, $ x^2 = \frac{28}{9} $, rationalized form, algebraic identity.\nMeta Description: Explore symbolic computation of quadratic roots, derive $ \frac{A + 2d}{A} $ from $ x^2 = \frac{28}{9} $, and understand root-product relationships using exact algebra."]