Multiply by \(u\): \( u^2 - 2u + 1 = 0 \Rightarrow (u - 1)^2 = 0 \Rightarrow u = 1 \).

["Understanding "Multiply by (u): Solving ( u^2 - 2u + 1 = 0 )"\nMaster the Perfect Square Step and the Unique Solution ( u = 1 )", "Solve the quadratic equation ( u^2 - 2u + 1 = 0 ) with clarity and confidence using the powerful method of completing the square. This foundational algebraic technique elegantly demonstrates how ( (u - 1)^2 = 0 ) leads directly to the solution — a critical concept in algebra, calculus, and engineering. Discover step-by-step how multiplying, factoring, and applying square identities reveal that ( u = 1 ) is not just a solution, but the only real solution. Learn how this method applies broadly to quadratic equations and beyond.", "---", "### What Does "Multiply by (u)" Mean in This Context?", "At first glance, "multiply by (u)" might seem misleading. In fact, the equation ( u^2 - 2u + 1 = 0 ) is not directly multiplied by (u), but we rearrange and factor it by treating (u) as a common term. This manipulation mirrors solving quadratic equations efficiently using algebra’s core tools—completing the square, factoring, and analyzing roots.", "---", "### The Equation: ( u^2 - 2u + 1 = 0 )", "Here’s the original quadratic equation:\n[\nu^2 - 2u + 1 = 0\n]", "Notice this expression closely resembles a perfect square trinomial — a special form where ( (u - a)^2 = u^2 - 2au + a^2 ), identifiable by equal linear and constant coefficients (minus signs).", "---", "### Step 1: Recognize the Perfect Square", "Compare:\n[\nu^2 - 2u + 1 \quad \ ext{vs.} \quad (u - a)^2 = u^2 - 2au + a^2\n]", "Matching constants:\n- Coefficient of (u) is (-2ab = -2) → (a = 1) (since (-2 \cdot 1 \cdot u = -2u))\n- Constant term (a^2 = 1^2 = 1) ✓", "Therefore:\n[\nu^2 - 2u + 1 = (u - 1)^2\n]", "---", "### Step 2: Rewrite the Equation in Factored Form", "Substitute the square form:\n[\n(u - 1)^2 = 0\n]", "---", "### Step 3: Solve the Simplified Equation", "Take the square root of both sides:\n[\nu - 1 = 0\n]", "Hence:\n[\nu = 1\n]", "This shows only one solution, but with multiplicity two, meaning (u = 1) is a repeated root.", "---", "### Why This Method Works: The Power of Completing the Square", "The technique of rewriting (u^2 - 2u + 1) as ((u - 1)^2) reveals far more than a single number. It demonstrates how:\n- Perfect squares appear naturally in algebraic identities.\n- Equations can collapse into simpler forms without complicated division.\n- Repeated roots emerge clearly — a key insight in polynomial analysis, graphing, and quadratic functions.", "This method is also widely used in calculus (e.g., when finding critical points), physics (modeling parabolic motion), and computer graphics (quadratic curves).", "---", "### Key Takeaways", "| Concept | Explanation |\n|---------|-------------|\n| Perfect Square Trinomial | (a^2 - 2ab + b^2 = (a - b)^2) always squares a binomial. |\n| Factoring | Recognizing patterns avoids brute-force solving. |\n| Solving ((u - 1)^2 = 0) | Only (u = 1) satisfies the equation; multiplicity indicates tangency in graphs. |\n| Real-World Use | Models instantaneous changes, optimal points, and symmetric phenomena. |", "---", "### How to Verify the Solution", "Plug ( u = 1 ) back into the original equation:\n[\n(1)^2 - 2(1) + 1 = 1 - 2 + 1 = 0 \quad \ ext{✓}\n]", "Confirmed — ( u = 1 ) lies perfectly on the parabola’s vertex.", "---", "### Final Thoughts", "The equation ( u^2 - 2u + 1 = 0 ) may be short, but it’s a gateway into deeper algebraic fluency. "Multiply by (u)" symbolizes the clever algebraic manipulations — factoring, completing the square — that turn complexity into clarity. Understanding this process equips learners to tackle similar quadratics and appreciate the elegance of algebra’s hidden structures.", "Remember: When faced with a quadratic, ask: Can I complete the square or factor directly? Often, ( (u - a)^2 = 0 ) is your strongest ally — yielding not just a solution, but insight into the equation’s shape and behavior.", "---", "Further Reading:\n- Completing the Square for quadratics\n- Perfect Squares and Their Graphs\n- Applications of Quadratic Equations in Real World Models", "---", "Keywords:*\n( u^2 - 2u + 1 = 0 ), solve quadratic, factor perfect square, solve ( (u - 1)^2 = 0 ), algebraic identities, completing the square, repeated roots, quadratic equations, ( u = 1 ) solution, algebra tutorial, quadratic roots explained."]