\( x^2 - 4x + 3 = (x - 3)(x - 1) = 0 \).

["# Solving the Quadratic Equation ( x^2 - 4x + 3 = (x - 3)(x - 1) = 0 )", "Understanding how to solve quadratic equations is a fundamental skill in algebra, crucial for students, educators, and math enthusiasts alike. One classic example is solving ( x^2 - 4x + 3 = 0 ), which factors neatly as ( (x - 3)(x - 1) = 0 ). This guide breaks down the problem step-by-step, explores factoring as a solution method, and explains the timeline for finding roots using the zero-product property.", "## What Is the Equation ( x^2 - 4x + 3 = 0 )?", "The equation ( x^2 - 4x + 3 = 0 ) is a second-degree polynomial. Quadratic equations take the standard form ( ax^2 + bx + c = 0 ), where ( a ), ( b ), and ( c ) are constants and ( a <br/>\ne 0 ). Here, the coefficients are:\n- ( a = 1 )\n- ( b = -4 )\n- ( c = 3 )", "The factored form ( (x - 3)(x - 1) = 0 ) reveals the roots directly, but understanding the transformation from standard form to factored form and applying the proper solving techniques is essential for deeper comprehension.", "## Factorization: Turning ( x^2 - 4x + 3 ) into ( (x - 3)(x - 1) )", "Factoring involves expressing the quadratic as a product of two binomials. We seek two numbers that multiply to ( +3 ) (the constant term) and add up to ( -4 ) (the coefficient of ( x )).", "- The pair ( -3 ) and ( -1 ) satisfies this:\n - Multiply: ( (-3) \ imes (-1) = 3 )\n - Add: ( -3 + (-1) = -4 )", "Using these, the equation factors as:\n[\nx^2 - 4x + 3 = (x - 3)(x - 1) = 0\n]", "This factorization confirms the roots simply by setting each binomial factor equal to zero:\n[\nx - 3 = 0 \quad \ ext{or} \quad x - 1 = 0\n]", "Solving these gives:\n- ( x = 3 )\n- ( x = 1 )", "These are the values of ( x ) that make the original equation true.", "## Applying the Zero-Product Property", "The zero-product property is a powerful algebraic tool: If the product of two factors is zero, then at least one of the factors must be zero.\nGiven\n[\n(x - 3)(x - 1) = 0\n]\nthen either:\n[\nx - 3 = 0 \quad \ ext{or} \quad x - 1 = 0\n]", "Solving these confirms:\n- ( x = 3 )\n- ( x = 1 )", "This method avoids the need for complex formulas when factoring is straightforward, making it efficient and intuitive for simple quadratics.", "## Solving Without Factoring: Quadratic Formula Alternative", "Though factoring is clean here, it’s valuable to know how to solve any quadratic when factoring is not obvious. The quadratic formula solves ( ax^2 + bx + c = 0 ) using:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For ( x^2 - 4x + 3 = 0 ):\n- ( a = 1 ), ( b = -4 ), ( c = 3 )\n- Discriminant: ( b^2 - 4ac = (-4)^2 - 4(1)(3) = 16 - 12 = 4 )\n- Roots:\n[\nx = \frac{4 \pm \sqrt{4}}{2} = \frac{4 \pm 2}{2}\n\Rightarrow x = \frac{6}{2} = 3 \quad \ ext{and} \quad x = \frac{2}{2} = 1\n]", "Same results — confirming the correctness of both methods.", "## Why Is This Equation Important?", "Understanding simple quadratics like ( x^2 - 4x + 3 = 0 ) lays the foundation for more advanced algebra, including graphing parabolas, solving real-world optimization problems, and working with polynomial equations. Factoring teaches logical thinking and pattern recognition—skills transferrable across many mathematical domains.", "## Step-by-Step Summary", "1. Start with the equation: ( x^2 - 4x + 3 = 0 )\n2. Factor to reveal roots: ( (x - 3)(x - 1) = 0 )\n3. Apply the zero-product property: ( x - 3 = 0 ) or ( x - 1 = 0 )\n4. Solve for ( x ): ( x = 3 ) or ( x = 1 )\n5. Verify using the quadratic formula (optional): Result confirms ( x = 1 ) and ( x = 3 )", "---", "Whether you’re a student learning to solve quadratics or a learner seeking clarity on algebraic fundamentals, mastering factoring and the zero-product property is essential. Practice with simple equations like ( x^2 - 4x + 3 = 0 ) builds confidence and precision—key to success in higher mathematics.", "Keywords: ( x^2 - 4x + 3 = 0 ), factoring quadratic, zero-product property, solving quadratics, algebraic methods, quadratic solutions, algebra for beginners, factoring techniques, quadratic formula, roots of equations."]