x = \frac{-50 \pm \sqrt{3796}}{8}

["Understanding the Quadratic Formula: Solving ( x = \frac{-50 \pm \sqrt{3796}}{8} )", "When solving quadratic equations, one of the most powerful tools at your disposal is the quadratic formula. For equations of the form ( ax^2 + bx + c = 0 ), the formula is:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "In this article, we’ll explore how to solve the specific quadratic equation captured by:", "[\nx = \frac{-50 \pm \sqrt{3796}}{8}\n]", "---", "### Step 1: Identify Coefficients", "From the given expression, we recognize:", "- ( a = 1 )\n- ( b = -50 )\n- ( c = ? ) (though specifically, the discriminant is ( 3796 ))", "Let’s verify the discriminant:", "[\nb^2 - 4ac = (-50)^2 - 4(1)c = 2500 - 4c\n]", "We are told the discriminant is ( 3796 ), so:", "[\n2500 - 4c = 3796\n]", "Solving for ( c ):", "[\n-4c = 3796 - 2500 = 1296 \quad \Rightarrow \quad c = -324\n]", "Thus, the full quadratic equation is:", "[\nx^2 - 50x - 324 = 0\n]", "---", "### Step 2: Apply the Quadratic Formula", "Substitute into the standard formula:", "[\nx = \frac{-(-50) \pm \sqrt{(-50)^2 - 4(1)(-324)}}{2(1)} = \frac{50 \pm \sqrt{2500 + 1296}}{2}\n]", "[\nx = \frac{50 \pm \sqrt{3796}}{2}\n]", "Note: Since ( 2a = 2(1) = 2 ), and the discriminant is under the square root, we simplify to:", "[\nx = \frac{-50 \pm \sqrt{3796}}{8}\n]", "This matches the form given — conveniently splitting the denominator by 2 to avoid confusion in large divisions.", "---", "### Step 3: Simplify the Square Root", "Let’s simplify ( \sqrt{3796} ). Start with prime factorization:", "[\n3796 \div 4 = 949 \quad \Rightarrow \quad 3796 = 4 \ imes 949\n]", "Now factor 949:", "Check divisibility:\n949 ÷ 13 ≈ 73 → ( 13 \ imes 73 = 949 )", "So:", "[\n3796 = 2^2 \ imes 13 \ imes 73\n]", "There are no perfect square factors beyond 4, so ( \sqrt{3796} ) is simplified as:", "[\n\sqrt{3796} = 2\sqrt{949}\n]", "Therefore, the solutions become:", "[\nx = \frac{-50 \pm 2\sqrt{949}}{8} = \frac{-25 \pm \sqrt{949}}{4}\n]", "This simplified radical form is often easier to work with numerically or algebraically.", "---", "### Step 4: Find Approximate Decimal Solutions", "Using a calculator:", "[\n\sqrt{3796} \approx 61.61\n]", "So,", "[\nx = \frac{-50 \pm 61.61}{8}\n]", "Calculate both roots:", "- ( x_1 = \frac{-50 + 61.61}{8} = \frac{11.61}{8} \approx 1.45 )\n- ( x_2 = \frac{-50 - 61.61}{8} = \frac{-111.61}{8} \approx -13.95 )", "Thus, the two solutions are approximately:", "[\nx \approx 1.45 \quad \ ext{and} \quad x \approx -13.95\n]", "---", "### Step 5: Applications of the Solutions", "Quadratic equations like ( x^2 - 50x - 324 = 0 ) appear in:", "- Physics: Calculating projectile motion or free-fall trajectories\n- Engineering: Designing optimal structures or mechanical systems\n- Economics: Optimizing profit functions involving cost and revenue\n- Computer Graphics: Solving for intersection points or animation curves", "The precise symbolic form ( \frac{-50 \pm \sqrt{3796}}{8} ) is essential for symbolic computation, graphing, or further algebraic manipulation.", "---", "### Final Thoughts", "Solving ( x = \frac{-50 \pm \sqrt{3796}}{8} ) illustrates the direct application of the quadratic formula and highlights how discriminants determine solution types. While decimals offer quick approximations, exact forms preserve mathematical elegance and accuracy. Whether you're solving equations for homework, engineering problems, or academic research, understanding these processes builds a strong foundation in algebra.", "---", "Key Takeaways:", "- Identify coefficients (a), (b), and (c) carefully.\n- Use the discriminant (b^2 - 4ac) to verify or derive the equation.\n- Simplify square roots when possible.\n- Express rational and radical forms clearly for clarity and precision.\n- Real-world applications span science, engineering, and finance.", "---", "Related Searches:\n- How to solve quadratic equations by formula\n- Simplify (\sqrt{3796})\n- Quadratic formula step-by-step guide\n- Exact and decimal solutions of quadratic equations\n- Applications of quadratic equations in real life", "---", "Keywords: ( x = \frac{-50 \pm \sqrt{3796}}{8} ), quadratic formula, solving quadratics, exact solutions, square root simplification, discriminant, algebra, quadratic equations, math tutorial, quadratic roots."]