\( x = rac{ -35 \pm \sqrt{35^2 + 4 \cdot 2 \cdot 102} }{4} = rac{ -35 \pm \sqrt{1225 + 816} }{4} = rac{ -35 \pm \sqrt{2041} }{4} \)
["Solving the Quadratic Equation: A Step-by-Step Guide to Finding ( x = \dfrac{-35 \pm \sqrt{2041}}{4} )", "Quadratic equations form the backbone of algebra and appear frequently in science, engineering, and economics. Understanding how to solve them not only sharpens analytical skills but also helps in modeling real-world problems. In this article, we’ll explore the derivation and solution of the equation:", "[\nx = \dfrac{ -35 \pm \sqrt{35^2 + 4 \cdot 2 \cdot 102} }{4}\n]", "We’ll break down each step to clarify the process of solving quadratic equations using the quadratic formula.", "---", "### Understanding the Standard Quadratic Equation", "A general quadratic equation is written as:", "[\nax^2 + bx + c = 0\n]", "The solutions to this equation are given by the quadratic formula:", "[\nx = \dfrac{ -b \pm \sqrt{b^2 - 4ac} }{2a}\n]", "---", "### Rewriting the Given Equation", "The equation we are solving is:", "[\nx = \dfrac{ -35 \pm \sqrt{35^2 + 4 \cdot 2 \cdot 102} }{4}\n]", "We observe that this expression matches the quadratic formula in form, but note the denominator: while standard form uses (2a), here the denominator is (4). This implies the equation was rewritten as:", "[\nx = \dfrac{ -35 \pm \sqrt{D} }{4}\n]", "which corresponds to (a = 2), since:", "[\n2a = 4 \quad \Rightarrow \quad a = 2\n]", "From the coefficients:", "- (a = 2)\n- (b = -35)\n- (c = 102)", "---", "### Compute the Discriminant", "The discriminant (D) is given by:", "[\nD = b^2 - 4ac = (-35)^2 - 4 \cdot 2 \cdot 102 = 1225 - 816 = 1225 + 816 = 2041\n]", "So, the full solution becomes:", "[\nx = \dfrac{ -(-35) \pm \sqrt{2041} }{4} = \dfrac{35 \pm \sqrt{2041}}{4}\n]", "Note the correction: since (b = -35), then (-b = +35), which justifies the numerator ( \pm \sqrt{2041} ) over (4).", "---", "### Step-by-Step Derivation", "1. Identify coefficients:\n (a = 2), (b = -35), (c = 102)", "2. Compute discriminant:\n [\n D = b^2 - 4ac = (-35)^2 - 4(2)(102) = 1225 - 816 = 2041\n ]", "3. Plug into quadratic formula:\n [\n x = \dfrac{ -b \pm \sqrt{D} }{2a} = \dfrac{35 \pm \sqrt{2041}}{2 \cdot 2} = \dfrac{35 \pm \sqrt{2041}}{4}\n ]", "---", "### Approximating the Roots", "For practical applications, the square root (\sqrt{2041}) is useful to approximate:", "[\n\sqrt{2041} \approx 45.18\n]", "So the two solutions are approximately:", "[\nx_1 \approx \dfrac{35 + 45.18}{4} = \dfrac{80.18}{4} \approx 20.05\n]", "[\nx_2 \approx \dfrac{35 - 45.18}{4} = \dfrac{-10.18}{4} \approx -2.55\n]", "---", "### Why This Method Matters", "Solving quadratics using the quadratic formula ensures accuracy, especially when factoring is difficult or impossible. Mastering this technique empowers learners to tackle complex equations in physics (e.g., motion under gravity), finance (e.g., profit modeling), and geometry (e.g., conic sections).", "---", "### Key Takeaways", "- Always identify coefficients (a), (b), and (c) correctly from the quadratic form.\n- The discriminant (b^2 - 4ac) determines the nature of the roots (real, repeated, or complex).\n- The plus-minus symbol ( \pm ) reflects two possible solutions.\n- Simplifying square roots improves readability and usefulness in applied contexts.", "---", "### Final Answer", "[\n\boxed{\nx = \dfrac{ -35 \pm \sqrt{2041} }{4} \approx 20.05 \quad \ ext{and} \quad x \approx -2.55\n}\n]", "---", "### Further Reading", "- Learn how to factor quadratics when applicable.\n- Explore real-world problems solved using quadratic equations.\n- Practice simplifying radicals and working with irrational numbers.", "If you found this content helpful, share it with fellow learners and visit us for more detailed lessons in algebra and mathematics!"]
