A quadratic equation \( x^2 - 5x + 6 = 0 \) is given. Find the roots and determine their product.
["# Solving the Quadratic Equation ( x^2 - 5x + 6 = 0 ): Finding the Roots and Their Product", "Quadratic equations are fundamental in algebra, serving as essential building blocks for more advanced mathematical concepts. One of the most common forms is ( ax^2 + bx + c = 0 ), and understanding how to solve it—especially by factoring—provides a clear path to discovering its roots and analyzing their properties. In this article, we’ll explore the quadratic equation ( x^2 - 5x + 6 = 0 ), find its roots, and determine the product of those roots using simplified methods.", "## Understanding the Quadratic Equation", "The general form of a quadratic equation is:", "[\nax^2 + bx + c = 0\n]", "For the equation ( x^2 - 5x + 6 = 0 ), comparing with the general form gives:\n- ( a = 1 )\n- ( b = -5 )\n- ( c = 6 )", "The roots of this equation are the values of ( x ) that satisfy the equation, meaning they make the quadratic expression equal to zero.", "## Factoring the Quadratic Equation", "One efficient way to solve ( x^2 - 5x + 6 = 0 ) is by factoring. We look for two numbers that multiply to ( c = 6 ) and add up to ( b = -5 ).", "The pairs of factors of 6 are:\n- ( 1 ) and ( 6 ): ( 1 + 6 = 7 ) (too high)\n- ( 2 ) and ( 3 ): ( 2 + 3 = 5 ), but we need (-2) and (-3) because the middle term is negative and the sum is (-5).", "Thus, we rewrite the equation as:", "[\n(x - 2)(x - 3) = 0\n]", "Setting each factor equal to zero gives the solutions:", "[\nx - 2 = 0 \quad \Rightarrow \quad x = 2\n]\n[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "So, the roots of the equation are ( x = 2 ) and ( x = 3 ).", "## Finding the Product of the Roots", "Rather than solving for the roots individually, there’s a powerful shortcut: the product of the roots of a quadratic equation ( ax^2 + bx + c = 0 ) is given by the formula:", "[\n\ ext{Product of roots} = \frac{c}{a}\n]", "In our equation, ( a = 1 ) and ( c = 6 ), so:", "[\n\frac{c}{a} = \frac{6}{1} = 6\n]", "Alternatively, multiplying the roots we found:", "[\n2 \ imes 3 = 6\n]", "Both methods confirm the product of the roots is 6.", "## Why Is This Useful?", "Knowing the product of the roots without solving the equation directly offers several advantages:", "- It saves time, especially when factoring is complex.\n- It helps verify solutions easily.\n- It reveals intrinsic properties of the quadratic, such as its relationship to the constant term ( c ) and leading coefficient ( a ).", "Moreover, this rule applies universally to all quadratics, making it a cornerstone in solving equations and modeling real-world problems involving parabolas and motion.", "## Conclusion", "Solving ( x^2 - 5x + 6 = 0 ) by factoring reveals roots ( x = 2 ) and ( x = 3 ). Their product is ( 2 \ imes 3 = 6 ), consistent with the algebraic identity ( \frac{c}{a} ). Mastering factoring and root-product relationships empowers students to tackle quadratic equations confidently, deepening their understanding of algebraic structures and preparing them for advanced topics in mathematics.", "Whether you're a student, teacher, or self-learner, mastering this classic equation equips you with essential skills for algebra and beyond."]
