Calculez : \( x = \frac{-50 \pm \sqrt{2500 + 1824}}{8} = \frac{-50 \pm \sqrt{4324}}{8} \).
["Title: Solving the Quadratic Equation: Step-by-Step Calculation of ( x = \frac{-50 \pm \sqrt{4324}}{8} )", "---", "### Understanding and Solving the Quadratic Equation", "The quadratic equation is a fundamental tool in algebra, widely used in science, engineering, and economics. Today, we’ll explore the precise calculation of:", "[\nx = \frac{-50 \pm \sqrt{2500 + 1824}}{8} = \frac{-50 \pm \sqrt{4324}}{8}\n]", "This equation stems from standard quadratic formulas applied to quadratic expressions, and solving it involves simplifying the radical and simplifying the expression fully.", "---", "### Step 1: Simplify the Expression Under the Square Root", "Start by computing the expression inside the square root:", "[\n2500 + 1824 = 4324\n]", "Thus, the equation becomes:", "[\nx = \frac{-50 \pm \sqrt{4324}}{8}\n]", "---", "### Step 2: Simplify ( \sqrt{4324} )", "To simplify ( \sqrt{4324} ), we look for perfect square factors.", "Factor ( 4324 ):", "- Divide by 4: ( 4324 \div 4 = 1081 )\n- So, ( 4324 = 4 \ imes 1081 )", "Now, check whether 1081 is a prime or contains a square factor:", "- Test divisibility:\n - Not divisible by 2, 3, 5, 7, 11, 13, 17, or 19 (all primes ≤ √1081 ≈ 32.9)\n - 1081 ends in 1, and checks confirm 1081 is prime", "Therefore, ( 4324 = 2^2 \ imes 1081 ), and:", "[\n\sqrt{4324} = \sqrt{4 \ imes 1081} = \sqrt{4} \ imes \sqrt{1081} = 2\sqrt{1081}\n]", "So now the expression becomes:", "[\nx = \frac{-50 \pm 2\sqrt{1081}}{8}\n]", "---", "### Step 3: Simplify the Fraction", "Factor numerator and denominator by 2:", "[\nx = \frac{2(-25 \pm \sqrt{1081})}{8} = \frac{-25 \pm \sqrt{1081}}{4}\n]", "This form is simplified and more readable.", "---", "### Final Answer", "[\n\boxed{x = \frac{-25 \pm \sqrt{1081}}{4}}\n]", "---", "### Why This Form Matters", "- The discriminant ( \Delta = 4324 ) determines the nature of the solutions. Since ( 1081 ) is not a perfect square, the roots are irrational and real.\n- Using exact radicals ensures precision in advanced applications such as calculus, physics modeling, and numerical analysis.\n- Simplified forms reduce computational errors and enhance readability for further mathematical reasoning.", "---", "### Summary", "To solve ( x = \frac{-50 \pm \sqrt{2500 + 1824}}{8} ):", "1. Calculate ( 2500 + 1824 = 4324 )\n2. Factor ( \sqrt{4324} = 2\sqrt{1081} )\n3. Simplify the fraction to ( \frac{-25 \pm \sqrt{1081}}{4} )", "Mastering such simplifications empowers you with greater control over quadratic solutions and opens doors to deeper mathematical mastery.", "---", "Keywords: quadratic equation solution, solve ( x = \frac{-50 \pm \sqrt{4324}}{8} ), simplify square roots, simplified radical form, $ x = \frac{-25 \pm \sqrt{1081}}{4} $, discriminant, irrational roots, algebraic expressions."]
