Solution: Let $ p(x) = ax^2 + bx + c $. Using the given values:

["# Solving Quadratic Equations: A Step-by-Step Guide Using $ p(x) = ax^2 + bx + c $", "Understanding quadratic equations is essential in algebra, and solving $ p(x) = ax^2 + bx + c $ is a foundational skill for students, engineers, and data analysts alike. Whether you're modeling real-world problems or optimizing algorithms, knowing how to solve quadratic equations sets the stage for advanced mathematical thinking. In this article, we’ll explore the standard solution method, discuss how given values impact the process, and provide a clear, practical approach to solving quadratic equations.", "## What is a Quadratic Function?", "A quadratic function is any polynomial of degree 2, expressed in the general form:\n$$\np(x) = ax^2 + bx + c\n$$\nwhere $ a, b, $ and $ c $ are constants, and $ a <br/>\neq 0 $. The graph of this function is a parabola, which can open upward (if $ a > 0 $) or downward (if $ a < 0 $). The solutions to $ p(x) = 0 $ correspond to the x-intercepts of the parabola — also known as the roots or zeros of the quadratic.", "---", "## The Quadratic Formula: A Universal Solution", "To solve $ ax^2 + bx + c = 0 $, the most reliable method is the quadratic formula:\n$$\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$\nThis formula works for any quadratic equation, regardless of whether it factors nicely or not.", "### Step-by-Step: How to Use the Quadratic Formula", "1. Identify coefficients: From $ p(x) = ax^2 + bx + c $, extract $ a $, $ b $, and $ c $.\n2. Compute the discriminant:\n $$\n \Delta = b^2 - 4ac\n $$\n The value of $ \Delta $ determines the nature of the roots:\n - If $ \Delta > 0 $: Two distinct real roots\n - If $ \Delta = 0 $: One real root (a repeated root)\n - If $ \Delta < 0 $: Two complex conjugate roots\n3. Apply the quadratic formula: Plug $ a $, $ b $, $ c $, and $ \Delta $ into the formula.\n4. Simplify to find $ x $: Calculate both roots using $ + $ and $ - $ signs.", "---", "## Using Given Values to Solve", "Let’s walk through a typical example to illustrate the process using actual values. Suppose we are given:\n$$\np(x) = 2x^2 - 5x + 2\n$$\nWe want to solve $ p(x) = 0 $.", "### Step 1: Identify coefficients\nHere, $ a = 2 $, $ b = -5 $, $ c = 2 $", "### Step 2: Compute the discriminant\n$$\n\Delta = (-5)^2 - 4(2)(2) = 25 - 16 = 9\n$$\nSince $ \Delta = 9 > 0 $, there are two distinct real roots.", "### Step 3: Apply the quadratic formula\n$$\nx = \frac{-(-5) \pm \sqrt{9}}{2(2)} = \frac{5 \pm 3}{4}\n$$", "### Step 4: Compute the roots\n$$\nx_1 = \frac{5 + 3}{4} = \frac{8}{4} = 2\n$$\n$$\nx_2 = \frac{5 - 3}{4} = \frac{2}{4} = \frac{1}{2}\n$$", "✅ So the solutions are $ x = 2 $ and $ x = \frac{1}{2} $. These roots correspond to where the parabola intersects the x-axis.", "---", "## Why Understanding This Matters", "Quadratic equations appear in various fields:\n- Physics: Motion under gravity (projectile motion)\n- Engineering: Optimization in design and cost modeling\n- Economics: Supply and demand curves often modeled quadratically\n- Computer Science: Algorithm efficiency analysis and mathematical optimization", "Mastering the solution of $ ax^2 + bx + c = 0 $ equips learners with a powerful tool for both academic problems and real-world applications.", "---", "## Summary", "- The general form of a quadratic is $ p(x) = ax^2 + bx + c $\n- Use the quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ to find all roots\n- The discriminant determines root nature—real and distinct, real and repeated, or complex\n- Practical examples using sample coefficients demonstrate clear, reproducible steps\n- Strong quadratic-solving skills enhance problem-solving across STEM disciplines", "---", "Try It Yourself:\nPick another quadratic from online problem sets or textbook exercises and walk through the steps above — you’ll quickly gain confidence in handling any quadratic equation.", "For deeper learning, explore how the discriminant influences graph behavior and how completing the square offers alternative insights. Mastering quadratics opens the door to advanced algebra and calculus concepts.", "---", "Happy Learning!\nUnderstanding quadratics is not just about equations — it’s about mastering a universal pattern in mathematics. Start solving today!"]