\( x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-94.5)}}{2} = \frac{-6 \pm \sqrt{36 + 378}}{2} = \frac{-6 \pm \sqrt{414}}{2} \)
["# Solving a Quadratic Equation: Step-by-Step Guide to ( x = \frac{-6 \pm \sqrt{414}}{2} )", "Solving quadratic equations is a fundamental skill in algebra and essential for applications across science, engineering, and finance. In this article, we focus on solving the quadratic equation:", "[\nx = \frac{-6 \pm \sqrt{6^2 - 4(1)(-94.5)}}{2} = \frac{-6 \pm \sqrt{414}}{2}\n]", "This equation arises when applying the quadratic formula to:", "[\nax^2 + bx + c = 0\n]", "where ( a = 1 ), ( b = -6 ), and ( c = -94.5 ). Let’s break down the process of solving this equation step-by-step and understand how to interpret the result.", "---", "## Understanding the Quadratic Formula", "The quadratic formula states:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plugging in ( a = 1 ), ( b = -6 ), and ( c = -94.5 ), we compute the discriminant:", "[\n\Delta = b^2 - 4ac = (-6)^2 - 4(1)(-94.5) = 36 + 378 = 414\n]", "The discriminant is positive (( \Delta = 414 > 0 )), which means the equation has two distinct real solutions.", "---", "## Step-by-Step Derivation", "Start with the quadratic expression:", "[\nx^2 - 6x - 94.5 = 0\n]", "Apply the quadratic formula:", "[\nx = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-94.5)}}{2(1)} = \frac{6 \pm \sqrt{36 + 378}}{2}\n]", "Simplify the square root:", "[\n\sqrt{414}\n]", "Thus, the full solution is:", "[\nx = \frac{6 \pm \sqrt{414}}{2}\n]", "Note that because ( b = -6 ), we have ( -b = 6 ), hence the simplified expression above.", "---", "## Simplifying the Root ( \sqrt{414} )", "We can simplify ( \sqrt{414} ) by factoring:", "[\n414 = 9 \ imes 46 = 3^2 \ imes 46\n]", "So:", "[\n\sqrt{414} = \sqrt{9 \cdot 46} = 3\sqrt{46}\n]", "Substituting back:", "[\nx = \frac{6 \pm 3\sqrt{46}}{2} = \frac{-6 \pm \sqrt{414}}{2} = \frac{6 \pm 3\sqrt{46}}{2}\n]", "This gives two solutions:", "[\nx_1 = \frac{6 + 3\sqrt{46}}{2}, \quad x_2 = \frac{6 - 3\sqrt{46}}{2}\n]", "---", "## Analyzing the Solutions", "Both solutions are irrational due to ( \sqrt{46} ), approximately ( 6.78 ). Estimating:", "- ( x_1 \approx \frac{6 + 3(6.78)}{2} = \frac{6 + 20.34}{2} = 13.17 )\n- ( x_2 \approx \frac{6 - 20.34}{2} = \frac{-14.34}{2} = -7.17 )", "The values illustrate how quadratic solutions often fall outside integers, especially when the discriminant is not a perfect square.", "---", "## Why This Equation Matters", "Quadratic equations model real-world phenomena like projectile motion, revenue optimization, and electrical resistance in circuits. Understanding how to solve them—especially when using the quadratic formula with non-integer constants—helps in accurate mathematical modeling and data analysis.", "---", "## Final Notes", "- The discriminant determines the nature of roots: positive for two real solutions, zero for one, and negative for complex ones.\n- Simplifying radicals enhances clarity and aids in interpreting exact vs. approximate values.\n- This method applies universally and forms the basis for solving higher-degree polynomials through factoring or numerical methods.", "Mastering quadratic equations like ( x = \frac{-6 \pm \sqrt{414}}{2} ) strengthens algebraic intuition and problem-solving precision.", "---", "### Want to practice more? Try solving:", "[\nx = \frac{4 \pm \sqrt{50 - 4(3)(7)}}{6} \quad \ ext{or} \quad x^2 + 5x + 12 = 0\n]", "---", "Keywords:\nquadratic equation solution, quadratic formula derivation, solve ( x = \frac{-6 \pm \sqrt{414}}{2} ), discriminant analysis, exact vs decimal solutions, algebra practice, real roots of quadratics"]
