A quadratic equation \(x^2 - 5x + 6 = 0\) is given. Find the roots using factoring.
["Solving the Quadratic Equation (x^2 - 5x + 6 = 0) by Factoring", "Quadratic equations are fundamental in algebra, serving as building blocks for understanding more complex mathematical models. One of the most efficient methods to solve quadratics is factoring, especially when the equation has whole-number roots. In this article, we’ll walk through how to find the roots of the quadratic equation (x^2 - 5x + 6 = 0) using factoring.", "---", "### What is a Quadratic Equation?", "A quadratic equation is a second-degree polynomial of the form:\n[\nax^2 + bx + c = 0\n]\nwhere (a), (b), and (c) are constants and (a <br/>\ne 0). In our case:\n[\nx^2 - 5x + 6 = 0\n]\nHere, (a = 1), (b = -5), and (c = 6).", "---", "### Why Use Factoring to Solve Quadratics?", "Factoring involves expressing the quadratic expression as a product of two binomials. If the expression can be written as ((x - p)(x - q) = 0), then the solutions (roots) are (x = p) and (x = q). This method is especially fast and clean when the quadratic factors nicely into integers.", "---", "### Step-by-Step Factorization of (x^2 - 5x + 6)", "We want to rewrite the equation (x^2 - 5x + 6 = 0) in factored form.", "1. Look for two numbers that multiply to (c = 6) (the constant term) and add to (b = -5) (the coefficient of the linear term).", "2. Think of pairs of integers that multiply to 6:\n - (1 \ imes 6 = 6), sum = 7\n - (2 \ imes 3 = 6), sum = 5\n - (-2 \ imes -3 = 6), sum = (-5) ✅", "3. The correct pair is (-2) and (-3) because they multiply to 6 and add to (-5).", "So, we factor the equation as:\n[\nx^2 - 5x + 6 = (x - 2)(x - 3) = 0\n]", "---", "### Solving for (x) Using the Factored Form", "Set each factor equal to zero:\n[\nx - 2 = 0 \quad \Rightarrow \quad x = 2\n]\n[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "Therefore, the solutions (roots) of the equation (x^2 - 5x + 6 = 0) are:\n[\n\boxed{x = 2} \quad \ ext{and} \quad \boxed{x = 3}\n]", "---", "### Verification Using the Zero Product Property", "Since ((x - 2)(x - 3) = 0), the product is zero when any factor is zero. Confirming:\n- (2 - 2 = 0) → valid\n- (3 - 3 = 0) → valid", "Thus, both solutions are correct.", "---", "### Why Learning Factoring Matters", "Mastering factoring helps in faster equation solving, proofs, and preparing for advanced math topics such as factoring higher-degree polynomials, solving systems, and understanding graph behavior. Practicing with simple quadratics like this lays a strong foundation for algebra mastery.", "---", "Summary:\nTo solve (x^2 - 5x + 6 = 0) by factoring:\n- Find two numbers multiplying to 6 and adding to (-5): (-2) and (-3).\n- Factor as ((x - 2)(x - 3) = 0).\n- Solve for (x): (x = 2) and (x = 3).", "This method offers clarity, speed, and concise answers when applicable.", "---", "Keywords: quadratic equation, solving (x^2 - 5x + 6 = 0), factoring quadratics, finding roots by factoring, algebra tutorial, quadratic formula shortcut, roots of equation, solving quadratics by factoring."]
