Question: A linguist’s model for word frequency is $ p(x) = x^2 - 2x + m $. Find the largest $ x $ such that $ p(x) = 0 $ when $ m = -3 $.

["Title: Solving $ p(x) = x^2 - 2x - 3 = 0 $: Find the Largest Root Using a Linguist’s Statistical Model", "Meta Description:\nDiscover how a linguist’s word frequency model $ p(x) = x^2 - 2x + m $ helps solve quadratic equations. Learn how to find the largest root when $ m = -3 $, with step-by-step algebra and language of linguistics in mathematics.", "---", "Introduction\nIn computational linguistics, modeling word frequency patterns helps uncover linguistic regularities. One useful mathematical tool in such models is a quadratic equation, such as $ p(x) = x^2 - 2x + m $, where $ x $ may represent word position, frequency rank, or some linguistic variable. When word frequency peaks or stabilizes at a critical value $ m = -3 $, determining the largest possible $ x $ satisfying $ p(x) = 0 $ reveals meaningful linguistic insights.", "In this article, we solve $ p(x) = x^2 - 2x - 3 = 0 $ using algebraic methods inspired by linguistic modeling, and identify the largest root — a key data threshold in frequency analysis.", "---", "Understanding the Model\nLinguists sometimes formalize word occurrence patterns using quadratic functions. Here, $ p(x) = x^2 - 2x + m $ represents a simplified frequency profile, where $ m $ adjusts the model to real-world data like Zipf’s law approximations. When analyzing linguistic variation or word rank frequencies, solving $ p(x) = 0 $ helps identify critical values where linguistic thresholds shift.", "Given:\n[\np(x) = x^2 - 2x + m\n]\nSet $ m = -3 $:\n[\np(x) = x^2 - 2x - 3\n]\nWe seek the largest real solution to $ x^2 - 2x - 3 = 0 $.", "---", "Step 1: Apply the Quadratic Formula\nThe general form $ ax^2 + bx + c = 0 $ yields solutions:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nFor $ x^2 - 2x - 3 = 0 $, coefficients are:\n- $ a = 1 $\n- $ b = -2 $\n- $ c = -3 $", "Plug in values:\n[\nx = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)} = \frac{2 \pm \sqrt{4 + 12}}{2} = \frac{2 \pm \sqrt{16}}{2}\n]\n[\nx = \frac{2 \pm 4}{2}\n]", "---", "Step 2: Compute the Roots\n[\nx_1 = \frac{2 + 4}{2} = \frac{6}{2} = 3\n]\n[\nx_2 = \frac{2 - 4}{2} = \frac{-2}{2} = -1\n]", "---", "Step 3: Identify the Largest Root\nThe two real solutions are $ x = 3 $ and $ x = -1 $. In linguistic modeling contexts—especially when $ x $ represents word position, rank, or a discrete variable—only non-negative values are meaningful. Thus, the largest valid solution is:\n[\n\boxed{x = 3}\n]", "This root may indicate a critical frequency threshold or a key rank in a linguistic corpus analyzed via quadratic modeling.", "---", "Conclusion\nUsing a linguist’s statistical model, solving $ p(x) = x^2 - 2x - 3 = 0 $ yields roots $ x = -1 $ and $ x = 3 $. The largest meaningful solution, $ x = 3 $, exemplifies how mathematical modeling supports linguistic analysis. By understanding such models, researchers can uncover patterns in word frequency distributions and predict linguistic behavior across corpora.", "Whether studying lexical frequency, sentence complexity, or discourse structure, quadratic equations remain powerful tools in computational linguistics — and solving $ p(x) = 0 $ reveals pivotal insights, starting with finding the largest $ x $.", "---", "Key Takeaways:\n- The largest solution to $ x^2 - 2x - 3 = 0 $ is $ x = 3 $.\n- Linguistic models use quadratic equations to analyze word frequency thresholds.\n- Solving for zero-crossings helps identify critical data points in language studies.", "Keywords: linguist’s model, word frequency, quadratic equation, $ p(x) = x^2 - 2x + m $, solve $ p(x) = 0 $, largest root, computational linguistics, mathematical modeling, linguistic analysis."]