Calculate: \(x = \frac{-50 \pm \sqrt{2500 + 2176}}{8} = \frac{-50 \pm \sqrt{4676}}{8}\).
["Title: How to Calculate ( x = \frac{-50 \pm \sqrt{2500 + 2176}}{8} ) – Step-by-Step Explanation", "When solving quadratic equations, one common challenge is simplifying complex expressions under the square root. This article breaks down the calculation:", "[\nx = \frac{-50 \pm \sqrt{2500 + 2176}}{8}\n]", "We'll guide you step-by-step through simplifying the expression and evaluating ( x ), featuring practical SEO keywords like “solve quadratic equation,” “simplify square root expression,” and “quadratic formula steps.”", "---", "### Step 1: Simplify the Expression Inside the Square Root", "Start by simplifying the expression under the square root:", "[\n2500 + 2176 = 4676\n]", "So the expression becomes:", "[\nx = \frac{-50 \pm \sqrt{4676}}{8}\n]", "---", "### Step 2: Simplify ( \sqrt{4676} )", "To simplify ( \sqrt{4676} ), factor the number into prime components:", "- Begin by checking divisibility by 4 (since 4676 ends in 6, and is divisible by 4):\n [\n 4676 \div 4 = 1169\n ]\n- So:\n [\n \sqrt{4676} = \sqrt{4 \ imes 1169} = 2\sqrt{1169}\n ]", "Now, check if 1169 is a perfect square or can be factored further:", "- Test divisibility: 1169 is not divisible by 2, 3, 5, 7, 11, or 13.\n- Test up to ( \sqrt{1169} \approx 34.2 ): 1169 is a prime number.", "Therefore:\n[\n\sqrt{4676} = 2\sqrt{1169}\n]", "---", "### Step 3: Substitute Back into the Formula", "Replace the square root in the original equation:", "[\nx = \frac{-50 \pm 2\sqrt{1169}}{8}\n]", "---", "### Step 4: Simplify the Fraction", "Factor numerator and denominator:", "[\nx = \frac{2(-25 \pm \sqrt{1169})}{8} = \frac{-25 \pm \sqrt{1169}}{4}\n]", "This is the fully simplified form.", "---", "### Final Answer:", "[\nx = \frac{-50 \pm \sqrt{4676}}{8} = \frac{-50 \pm 2\sqrt{1169}}{8} = \frac{-25 \pm \sqrt{1169}}{4}\n]", "---", "### Why This Simplification Matters", "Understanding how to simplify square roots in quadratic equations improves clarity and precision in algebra. It supports better problem-solving when applying the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Feel free to use this guide when tackling equations with large or non-perfect square constants—breaking the steps down makes solving easier and errors fewer.", "---", "Keywords:\n- Solve quadratic equation\n- Simplify square root expression\n- Quadratic formula steps\n- Simplify ( \sqrt{4676} )\n- Rationalize quadratic expressions\n- Algebraic simplification guide", "Meta Title:\nCalculate ( x = \frac{-50 \pm \sqrt{2500 + 2176}}{8} ): Step-by-step simplification and solution", "Meta Description:\nLearn how to fully simplify ( x = \frac{-50 \pm \sqrt{2500 + 2176}}{8} ) by evaluating inside-the-root terms, simplifying the radical, and applying algebraic steps for the quadratic formula. Perfect for algebra students and problem solvers."]
