\[ x = \frac{9 \pm \sqrt{(-9)^2 - 4 \times 2 \times 7}}{2 \times 2} = \frac{9 \pm \sqrt{81 - 56}}{4} = \frac{9 \pm \sqrt{25}}{4} = \frac{9 \pm 5}{4}. \]

["Understanding the Quadratic Formula: Simplifying the Solution to ( x = \frac{9 \pm \sqrt{25}}{4} )", "Solve quadratic equations effortlessly with the quadratic formula — a powerful tool used across mathematics, engineering, and physics. One concise yet elegant example starts with:", "[\nx = \frac{9 \pm \sqrt{(-9)^2 - 4 \ imes 2 \ imes 7}}{2 \ imes 2} = \frac{9 \pm \sqrt{81 - 56}}{4} = \frac{9 \pm \sqrt{25}}{4} = \frac{9 \pm 5}{4}\n]", "This equation simplifies the solution process for any quadratic in the form ( ax^2 + bx + c = 0 ), showcasing how substitution and simplification lead to clear, concise answers.", "---", "### What Is the Quadratic Formula?", "The quadratic formula provides the solutions to any standard quadratic equation:", "[\nax^2 + bx + c = 0 \quad \ ext{where} \quad a <br/>\ne 0\n]", "The solution is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Standardizing expressions helps improve clarity and avoids computational errors, especially in complex calculations.", "---", "### Step-by-Step Simplification of ( x = \frac{9 \pm \sqrt{25}}{4} )", "Given the discriminant:\n[\n(-9)^2 - 4 \ imes 2 \ imes 7 = 81 - 56 = 25\n]", "The square root simplifies directly:", "[\n\sqrt{25} = 5\n]", "Substituting back gives:", "[\nx = \frac{9 \pm 5}{4}\n]", "This reveals two possible solutions:", "- ( x = \frac{9 + 5}{4} = \frac{14}{4} = 3.5 )\n- ( x = \frac{9 - 5}{4} = \frac{4}{4} = 1 )", "These roots correspond to the points where the parabola ( y = 2x^2 + 9x + 7 ) crosses the x-axis.", "---", "### Why Simplify Quadratic Solutions?", "- Clarity: Simplified expressions are easier to interpret and use in further calculations.\n- Efficiency: Standardized forms reduce errors in hand calculations and programming implementations.\n- Visual Appeal: Clean expressions improve readability in textbooks, reports, and presentations.", "---", "### Applications of Quadratic Solutions", "From physics — solving motion under gravity — to economics modeling profit maximization, quadratic equations underpin many real-world problems. Efficiently solving for ( x ) empowers accurate predictions and insights.", "---", "### Key Takeaways", "- The quadratic formula transforms messy equations into clear, solvable forms.\n- Simplifying radicals like ( \sqrt{25} ) improves readability and accuracy.\n- Solved expressions often reveal critical values and root behavior.\n- Understanding this structure builds a foundation for advanced algebra and calculus.", "---", "Explore more about how the quadratic formula executes in various fields and practice solving equations step by step — master this skill today!"]