\[ x^2 - 4x + 3 = (x - 1)(x - 3). \]

["Understanding the Factorization of ( x^2 - 4x + 3 = (x - 1)(x - 3) )", "Solving quadratic equations can often be simplified using factoring. One memorable identity you may encounter is:", "[\nx^2 - 4x + 3 = (x - 1)(x - 3)\n]", "### What Does the Equation Represent?", "The expression on the left, ( x^2 - 4x + 3 ), is a quadratic trinomial. Factoring it over the integers reveals two binomial factors: ( (x - 1) ) and ( (x - 3) ). This means:", "[\nx^2 - 4x + 3 = (x - 1)(x - 3)\n]", "### Quick Proof via Expansion", "To verify the factorization, expand the right-hand side:", "[\n(x - 1)(x - 3) = x \cdot x + x \cdot (-3) + (-1) \cdot x + (-1) \cdot (-3)\n]\n[\n= x^2 - 3x - x + 3 = x^2 - 4x + 3\n]", "The result matches the original quadratic, confirming the accuracy of the factorization.", "### Solving the Equation", "This factorization makes solving ( x^2 - 4x + 3 = 0 ) straightforward. Setting each factor equal to zero gives:", "[\nx - 1 = 0 \quad \Rightarrow \quad x = 1\n]\n[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "So, the solutions are ( x = 1 ) and ( x = 3 ).", "### Why Is This Factorization Useful?", "- Simplifying expressions: Breaking polynomials into factors helps in simplifying rational expressions and integration in calculus.\n- Solving equations faster: Recognizing factored forms allows quick identification of solutions.\n- Understanding roots: Each factor ( x - c ) indicates a root at ( x = c ), making the graph’s behavior intuitive.", "### Final Thoughts", "The identity\n[\nx^2 - 4x + 3 = (x - 1)(x - 3)\n]\nexemplifies how quadratic expressions can be rewritten for deeper insight and easier manipulation. Whether in algebra, calculus, or application-based problem solving, mastering such factorizations is essential for students and educators alike.", "Mastering expressions like this strengthens your algebraic toolkit—essential for advanced math studies and real-world applications.", "---", "Keywords:\n( x^2 - 4x + 3 ), factoring quadratic, factorization example, solve quadratic equation, expand ( (x - 1)(x - 3) ), algebraic identity, polynomial decomposition, root finding quadratics."]