So: \( x = \frac{35 - \sqrt{985}}{4} \) (take smaller root since larger would go negative).
["Smallest Real Root: Simplifying the Quadratic Solution ( x = \frac{35 - \sqrt{985}}{4} )", "When solving quadratic equations, one common challenge is correctly identifying the smallest (or most negative) real root—especially when the quadratic formula yields two roots, one positive and one negative. In the case of the expression\n[\nx = \frac{35 - \sqrt{985}}{4},\n]\nwe focus on the smaller root due to the constraint that the larger root becomes negative, making it unsuitable in contexts requiring non-negative solutions.", "### Understanding the Quadratic Equation", "This expression arises from solving a quadratic equation of the form:\n[\nax^2 + bx + c = 0,\n]\nwhere the discriminant ( D = b^2 - 4ac ) is positive, ensuring two real roots. Applying the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\n]\nwe select either the upper or lower sign depending on the desired root. Since we want the smaller root, choosing the negative sign gives:\n[\nx = \frac{-b - \sqrt{b^2 - 4ac}}{2a}.\n]\nFor our specific case, with ( a = 1 ), ( b = 0 ), and ( c = -\frac{35}{4} ) (derived from rearranging the root expression), this leads naturally to\n[\nx = \frac{35 - \sqrt{985}}{4}.\n]", "### Why Take the Smaller Root?", "The larger root\n[\nx = \frac{35 + \sqrt{985}}{4}\n]\nevaluates to a value greater than 10, whereas the smaller root\n[\nx \approx \frac{35 - 31.39}{4} \approx \frac{3.61}{4} \approx 0.9025\n]\nremains a small positive number—ideal for applications requiring non-negative values (e.g., physical measurements, financial metrics, or probability ranges).", "### Simplifying and Evaluating the Expression", "Let’s simplify and compute:\n( \sqrt{985} ) is an irrational number, approximately equal to 31.39 (using a calculator or known approximation). Thus,\n[\nx = \frac{35 - 31.392}{4} = \frac{3.608}{4} \approx 0.902.\n]\nThis small positive root emerges naturally from the quadratic’s structure, avoiding negative values entirely.", "### Practical Applications", "Such roots appear in:\n- Engineering tolerances: Where negative measurements have no physical meaning.\n- Financial models: Restricting loan amounts or profit thresholds to positive values.\n- Sciency computations: When modeling population growth constrained to grow from small positive initial values.", "### Final Thoughts", "Selecting the smaller root in quadratic equations is not only mathematically correct but strategically valuable—ensuring solutions align with real-world constraints. The expression ( x = \frac{35 - \sqrt{985}}{4} ) exemplifies how root selection impacts usability and accuracy in applied contexts. Always verify root signs when solving equations with real-world implications.", "---", "Key takeaway: For quadratics yielding a positive smaller root—especially when a larger root becomes negative—opting for the negative sign in the quadratic formula ensures not just mathematical correctness but practical relevance.", "Keywords: quadratic root selection, smaller positive root, ( \frac{35 - \sqrt{985}}{4} ), real roots analysis, negative vs positive roots, quadratic formula applications, non-negative solutions."]
