Set \( u + rac{1}{u} = 2 \). Multiply by \(u\): \( u^2 - 2u + 1 = 0 \Rightarrow (u - 1)^2 = 0 \Rightarrow u = 1 \), still.
["# Solving the Equation ( u + \frac{1}{u} = 2 ): A Deep Dive", "The equation ( u + \frac{1}{u} = 2 ) is a classic algebraic expression that appears frequently in mathematics education and forms the basis for understanding quadratic equations, identities, and rational functions. Despite its simplicity, solving it reveals key algebraic techniques and fundamental identities. This article explains how to solve the equation step-by-step, why multiplying through by ( u ) leads neatly to a clean solution, and the deeper meaning behind the result.", "---", "## Understanding the Equation", "We begin with:", "[\nu + \frac{1}{u} = 2\n]", "This equation involves ( u ) and its reciprocal ( \frac{1}{u} ), making it undefined when ( u = 0 ) due to division by zero. Thus, ( u <br/>\neq 0 ) is essential. Our goal is to solve for all valid real or complex values of ( u ) satisfying this relation.", "---", "## Eliminate the Fraction: Multiply Through by ( u )", "To simplify the equation and avoid working with fractions, the standard technique is to multiply both sides by ( u ), provided ( u <br/>\neq 0 ):", "[\nu \cdot \left( u + \frac{1}{u} \right) = 2u\n]", "Using the distributive property:", "[\nu^2 + 1 = 2u\n]", "Rearranging terms gives a standard quadratic equation:", "[\nu^2 - 2u + 1 = 0\n]", "This is a perfect square trinomial and can be factored as:", "[\n(u - 1)^2 = 0\n]", "---", "## Solve the Quadratic Equation", "[\n(u - 1)^2 = 0 \Rightarrow u - 1 = 0 \Rightarrow u = 1\n]", "The only solution is ( u = 1 ). Since the equation reduces to a perfect square, it is a double root — both roots are identical.", "---", "## Why This Solution Makes Sense", "At first glance, one might wonder: “Why isn’t there more than one solution, like for quadratic equations with two roots?” The reason lies in the domain restriction ( u <br/>\neq 0 ) and in the nature of the equation.", "Although the original equation excludes ( u = 0 ), it introduces no new solutions. The transformed quadratic correctly captures all possible values of ( u ) satisfying the original equation, with the sole solution being:", "[\n\boxed{u = 1}\n]", "If we test ( u = 1 ):", "[\n1 + \frac{1}{1} = 1 + 1 = 2\n]", "The solution checks perfectly.", "---", "## Mathematical Insight: Identity and Symmetry", "Beyond solving, the equation ( u + \frac{1}{u} = 2 ) reveals a deeper identity. When ( u + \frac{1}{u} = 2 ), we recognize this as the Arithmetic Mean – Geometric Mean (AM-GM) Equality in disguise.", "For any positive ( u ), the AM-GM inequality states:", "[\n\frac{u + \frac{1}{u}}{2} \geq \sqrt{u \cdot \frac{1}{u}} = 1\n]", "Equality occurs if and only if ( u = \frac{1}{u} \Rightarrow u^2 = 1 \Rightarrow u = \pm 1 ). But only ( u = 1 ) satisfies the original equation and avoids making denominators zero.", "Thus, the equation encapsulates both algebraic solution and a foundational mathematical inequality.", "---", "## Applications and Extensions", "This equation appears in:", "- Trigonometric identities (e.g., when solving ( x + \frac{1}{x} = 2 \cos \ heta ))\n- Complex analysis, where poles and residues relate to singularities at ( u = 0 )\n- Algebraic manipulation exercises to emphasize care with domain and equivalence", "It serves as a gateway to understanding rational functions, transformations, and symmetry in equations.", "---", "## Conclusion", "The equation ( u + \frac{1}{u} = 2 ) may seem simple, but its solution reveals elegant algebraic principles and ties into important mathematical concepts like AM-GM inequality and function identity. By multiplying through by ( u ), we avoid fractions, reduce the equation to a quadratic, and arrive cleanly at the unique solution:", "[\n\boxed{u = 1}\n]", "Always remember: ( u = 0 ) is excluded from the domain, ensuring the validity of the solution.", "---", "Understanding how such equations are solved strengthens foundational algebra skills and prepares learners for more advanced topics in mathematics and related fields.", "---", "Keywords: ( u + \frac{1}{u} = 2 ), solve equation, algebraic identity, quadratic formula, ( (u-1)^2 = 0 ), reciprocal equation, AM-GM inequality, rational functions, mathematical solution, single root, double root.", "---", "For further reading on solving rational equations and mastering algebraic identities, explore resources on quadratic reasoning and function comparison."]
