So exact answer is \( rac{ -35 + \sqrt{2041} }{4} \), but only positive root.
["# The Exact Positive Root: Why ( \frac{ -35 + \sqrt{2041} }{4} ) Is the Solution to Quadratic Equations", "When solving quadratic equations of the form ( ax^2 + bx + c = 0 ), one fundamental concept is identifying the correct root—especially when only the positive solution matters. For the specific equation where the exact solution is ( \frac{ -35 + \sqrt{2041} }{4} ), understanding why this is the only positive root delivers clarity and strengthens problem-solving skills in algebra.", "## Understanding the Quadratic Formula", "Recall that the quadratic formula provides two solutions for any quadratic equation:", "[\nx = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}\n]", "In our case, the quadratic expression evaluates to ( \frac{ -35 + \sqrt{2041} }{4} ), arising from substituting values into the formula and selecting the positive root.", "## Analyzing the Root Expression", "The full root calculation begins with:", "[\n\frac{ -35 \pm \sqrt{ (-35)^2 - 4 \cdot 1 \cdot c} }{2 \cdot 1}\n]", "Given the final simplified positive root ( \frac{ -35 + \sqrt{2041} }{4} ), it becomes clear:", "- The minus sign before the square root yields a smaller number but always risks a negative or possibly positive outcome depending on constants.\n- The plus sign would give ( \frac{ -35 + \sqrt{2041} }{4} ), which evaluates approximately to ( \frac{ -35 + 45.18 }{4} \approx \frac{10.18}{4} \approx 2.545 ), a positive value.", "The negative sign combined with the square root yields a smaller positive result, showing that only the positive root ( \frac{ -35 + \sqrt{2041} }{4} ) satisfies conditions such as positivity and physical relevance in many applications.", "## Why Focus on the Positive Root?", "In real-world contexts—such as finance, physics, or engineering—only positive solutions often make sense. For example, time, distance, and investment returns are non-negative by definition. Even if multiple mathematical roots exist, selecting the positive one ensures practical applicability.", "## Verifying the Discriminant and Solution Validity", "To confirm correctness, compute the discriminant:", "[\n\Delta = (-35)^2 - 4 \cdot 1 \cdot c = 1225 - 4c\n]", "For the given root to be real, ( \Delta \geq 0 \Rightarrow 2041 \geq 0 ), which holds true. Thus, the solution is not only mathematically sound but also applicable in contexts requiring real, and specifically positive, results.", "## Conclusion", "The exact positive root ( \frac{ -35 + \sqrt{2041} }{4} ) represents a precise answer to a quadratic equation, selected for its positive value and real-life relevance. Embracing this method enhances your ability to confidently determine and interpret solutions in algebra—ensuring accuracy and clarity whenever the positive outcome matters.", "Whether solving equations by hand or using calculators, remembering why this is the positive root strengthens fundamental understanding and builds confidence in mathematical problem-solving."]
