x = \frac{-42 \pm 6\sqrt{7}}{28} = \frac{-21 \pm 3\sqrt{7}}{14}.

["Understanding the Simplified Form of a Quadratic Equation: Solving ( x = \frac{-42 \pm 6\sqrt{7}}{28} ) and Its Simplified Version ( x = \frac{-21 \pm 3\sqrt{7}}{14} )", "When solving quadratic equations of the form ( ax^2 + bx + c = 0 ), survival often depends on simplifying complex expressions. One such instance is transforming the solution of a quadratic into its most reduced form. Consider the equation leading to:", "[\nx = \frac{-42 \pm 6\sqrt{7}}{28}\n]", "This expression arises when applying the quadratic formula to the equation ( x^2 + \left(\frac{21}{14}\right)x + \ ext{rational terms} = 0 ), but it’s not yet in its cleanest, most usable form. Let’s examine how this simplifies to:", "[\nx = \frac{-21 \pm 3\sqrt{7}}{14}\n]", "---", "### Why Simplify Quadratic Solutions?", "- Improved readability: Reduced fractions and like terms make the solution easier to interpret and apply in further calculations.\n- Easier computation: Working with smaller integers and simplified radicals reduces human error and streamlines arithmetic.\n- Standard practice: Educational and professional settings favor reduced expressions aligned with algebraic best practices.", "---", "### Step-by-Step Simplification", "Start with the original expression:", "[\nx = \frac{-42 \pm 6\sqrt{7}}{28}\n]", "Divide both the numerator and denominator by the greatest common divisor of 42, 6, and 28, which is 14:", "- Numerator: (-42 \div 14 = -3), ( \pm 6\sqrt{7} \div 14 = \pm \frac{6}{14}\sqrt{7} = \pm \frac{3}{7}\sqrt{7} ) — but actually divide simply:\n (-42 \div 14 = -3), (6\sqrt{7} \div 14 = \frac{3\sqrt{7}}{7} ). Wait — better split the division:", "Actually, fully:", "[\n\frac{-42}{28} = -\frac{3}{2}, \quad \frac{6\sqrt{7}}{28} = \frac{3\sqrt{7}}{14}\n]", "So:", "[\nx = -\frac{3}{2} \pm \frac{3\sqrt{7}}{14}\n]", "But to combine into a single fraction with denominator 14:", "Rewrite (-\frac{3}{2}) as (-\frac{21}{14}), and keep ( \frac{3\sqrt{7}}{14} ):", "[\nx = \frac{-21 \pm 3\sqrt{7}}{14}\n]", "---", "### Key Takeaways", "- Simplified form clarifies radical expressions and connects directly to standard quadratic solution forms.\n- Divisibility by common factors (here, 14) is key to reducing fractions cleanly.\n- Maintaining equivalent meaning ensures no loss of mathematical precision during simplification.", "---", "### Final Thoughts", "Mastering expression simplification—especially in quadratic solutions—enhances both problem-solving fluency and communication in mathematics. Remember:\n[\nx = \frac{-42 \pm 6\sqrt{7}}{28} = \frac{-21 \pm 3\sqrt{7}}{14}\n]", "is not just equivalent, but thoughtfully reduced for clarity and ease. Whether studying, teaching, or applying algebra, precision and simplicity go hand in hand.", "---", "Keywords: quadratic equation simplification, rationalizing radicals, solving quadratic formula, simplifying ( x = \frac{-42 \pm 6\sqrt{7}}{28} ), reducing fractions in radicals, algebra simplification, math problem solving, simplified quadratic solutions.", "---", "Need help with quadratic equations? Mastering simplification like this will strengthen your algebra foundation and boost your problem-solving confidence."]