But from quadratic: $ 14x^2 + 42x + 27 = 0 $. Product of roots $ \frac{27}{14} $, sum $ -3 $.
["Understanding the Quadratic Equation: $ 14x^2 + 42x + 27 = 0 $", "Quadratic equations are fundamental in algebra, offering powerful tools to model real-world problems and solve complex mathematical relationships. One such equation that exemplifies key properties of quadratics is:", "$$\n14x^2 + 42x + 27 = 0\n$$", "In this article, we’ll explore essential characteristics of this quadratic—specifically, the product of its roots and the sum of its roots—and verify these using the quadratic formula and fundamental algebra.", "---", "### Recalling the Standard Form", "A standard quadratic equation is written as:", "$$\nax^2 + bx + c = 0\n$$", "For the given equation:", "- $ a = 14 $\n- $ b = 42 $\n- $ c = 27 $", "---", "### Sum and Product of Roots: Basics", "For any quadratic equation $ ax^2 + bx + c = 0 $, the sum and product of the roots are given by:", "- Sum of roots: $ r_1 + r_2 = -\frac{b}{a} $\n- Product of roots: $ r_1 \cdot r_2 = \frac{c}{a} $", "---", "### Calculating the Sum of Roots", "Using the formula:", "$$\nr_1 + r_2 = -\frac{b}{a} = -\frac{42}{14} = -3\n$$", "Thus, the sum of the roots is $ -3 $. This confirms one key property without solving the equation explicitly.", "---", "### Calculating the Product of Roots", "Using the product formula:", "$$\nr_1 \cdot r_2 = \frac{c}{a} = \frac{27}{14}\n$$", "This tells us the product of the roots is $ \frac{27}{14} $. Interestingly, unlike equations with integer roots, here the roots multiply to a reduced fraction, reflecting a non-integer solution set.", "---", "### Verifying Results with the Quadratic Formula", "To ensure accuracy, compute the roots directly using the quadratic formula:", "$$\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$", "First, compute the discriminant:", "$$\n\Delta = b^2 - 4ac = 42^2 - 4 \cdot 14 \cdot 27 = 1764 - 1512 = 252\n$$", "Since $ \Delta = 252 = 36 \ imes 7 $, we write:", "$$\n\sqrt{252} = \sqrt{36 \cdot 7} = 6\sqrt{7}\n$$", "Now, substitute:", "$$\nx = \frac{-42 \pm 6\sqrt{7}}{2 \cdot 14} = \frac{-42 \pm 6\sqrt{7}}{28} = \frac{-21 \pm 3\sqrt{7}}{14}\n$$", "So the roots are:", "$$\nx_1 = \frac{-21 + 3\sqrt{7}}{14}, \quad x_2 = \frac{-21 - 3\sqrt{7}}{14}\n$$", "---", "### Sum of Roots from Solved Values", "Add the two roots:", "$$\nx_1 + x_2 = \frac{(-21 + 3\sqrt{7}) + (-21 - 3\sqrt{7})}{14} = \frac{-42}{14} = -3\n$$", "Consistent with the formula.", "---", "### Product of Roots from Solved Values", "Multiply:", "$$\nx_1 \cdot x_2 = \left( \frac{-21 + 3\sqrt{7}}{14} \right) \left( \frac{-21 - 3\sqrt{7}}{14} \right) = \frac{(-21)^2 - (3\sqrt{7})^2}{14^2}\n$$", "$$\n= \frac{441 - 9 \cdot 7}{196} = \frac{441 - 63}{196} = \frac{378}{196} = \frac{27}{14}\n$$", "Again, confirms the earlier result.", "---", "### Conclusion", "The quadratic equation $ 14x^2 + 42x + 27 = 0 $ illustrates classic properties of quadratic roots:", "- Sum of the roots: $ -3 $\n- Product of the roots: $ \frac{27}{14} $", "These values are verified both through the fundamental formulas and direct application of the quadratic formula. Understanding these relationships empowers students to solve quadratics efficiently and interpret their solutions meaningfully.", "Whether applying this to physics, economics, or engineering, mastering these concepts strengthens problem-solving skills and deepens mathematical intuition.", "---", "### Keywords for SEO Optimization", "- Quadratic equation solutions\n- Sum of roots quadratic formula\n- Product of roots algebra\n- Solve $ 14x^2 + 42x + 27 = 0 $\n- Quadratic properties and formulas\n- Algebraic verification with discriminant\n- Real roots of quadratic equations\n- Learning product and sum of roots", "---", "Explore these key ideas to build a solid foundation in quadratic equations and excel in algebra!"]
