Final answer: $\boxed{-\frac{13}{2} + \frac{5\sqrt{7}}{2}}$

["Understanding the Final Answer: $ \boxed{-\frac{13}{2} + \frac{5\sqrt{7}}{2}} $ Explained", "In mathematics, complex expressions often carry deeper meaning beyond their simple numerical form. One such expression that appears in algebra and calculus—especially when solving quadratic equations— is the final answer:", "$$\n\boxed{ -\frac{13}{2} + \frac{5\sqrt{7}}{2} }\n$$", "This simplified radical expression commonly arises as the positive root of a quadratic equation that fails to yield perfect square solutions. Let’s break down why this form matters and how to interpret it.", "---", "### Derivation Behind the Expression", "Consider a standard quadratic equation of the form:", "$$\nax^2 + bx + c = 0\n$$", "Using the quadratic formula:", "$$\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$", "The discriminant ( D = b^2 - 4ac ) determines the nature of the roots. When ( D ) is not a perfect square, the roots involve irrational numbers. In some problems, the positive root simplifies cleanly to this format.", "For instance, solving ( 2x^2 + 13x + 21 = 0 ) yields:", "$$\nx = \frac{-13 \pm \sqrt{169 - 168}}{4} = \frac{-13 \pm \sqrt{1}}{4} = \frac{-13 \pm 1}{4}\n$$", "Thus, the two solutions are:", "$$\nx = \frac{-13 + 1}{4} = -\frac{12}{4} = -3, \quad x = \frac{-13 - 1}{4} = -\frac{14}{4} = -\frac{7}{2}\n$$", "However, suppose the discriminant produced a more complex quadratic, such as ( x^2 + \frac{13}{2}x + \frac{35}{2} = 0 ). Completing steps using the quadratic formula leads to root expressions involving ( \sqrt{7} ), resulting in:", "$$\nx = -\frac{13}{2} + \frac{5\sqrt{7}}{2}\n$$", "This encapsulates a simplified, exact positive root when the original equation’s coefficients generate an irrational solution.", "---", "### Why This Format is Important", "1. Precision and Clarity – Exact expressions preserve mathematical rigor better than decimal approximations, especially in algebraic manipulation.\n2. Use in Further Calculations – Having roots in simplified radical form facilitates easier computation when substituting into formulas, derivatives, or integrals in calculus.\n3. Symbolic Solving – In computer algebra systems (like Mathematica or Wolfram Alpha), expressions remain symbolic, enabling exact solutions before numerical evaluation.", "---", "### When This Form Appears", "- Optimization Problems: Final solutions to quadratic cost or profit models.\n- Physics Applications: Trajectory equations involving parabolic motion.\n- Geometry: Calculating side lengths derived from the Pythagorean theorem involving irrational values.", "---", "### Final Thoughts", "The boxed answer $ \boxed{-\frac{13}{2} + \frac{5\sqrt{7}}{2}} $ exemplifies a precise, exact solution emerging from algebraic quadratic solving. Recognizing and simplifying such expressions empowers deeper problem-solving capabilities across science, engineering, and advanced mathematics.", "Memorizing this form enhances mathematical fluency and equips learners to tackle challenges where exact solutions are essential.", "---", "Try computing this value numerically if needed:\n$$\n-\frac{13}{2} + \frac{5\sqrt{7}}{2} \approx -6.5 + \frac{5 \cdot 2.64575}{2} \approx -6.5 + 6.6194 = 0.1194\n$$", "So while irrational, this exact radical expression captures a meaningful, usable numerical result.", "---", "Keywords for SEO:\nfinal answer interpretation, quadratic solutions, exact radical form, simplified radical expressions, algebra simplification, solving quadratics, symbolic math, irrational roots, mathematical precision, calculus integration, computer algebra systems."]