A quadratic equation \( x^2 - 5x + 6 = 0 \) needs to be solved. Find the roots.
["Finding the Roots of the Quadratic Equation ( x^2 - 5x + 6 = 0 )", "Solving quadratic equations is a fundamental skill in algebra, and it often appears in academic and practical applications. One commonly encountered quadratic equation is:", "[\nx^2 - 5x + 6 = 0\n]", "In this article, we will explore how to find the roots of this equation step-by-step using factoring, a straightforward and efficient method when applicable.", "---", "### What Is a Quadratic Equation?", "A quadratic equation is any equation of the form:", "[\nax^2 + bx + c = 0\n]", "where ( a ), ( b ), and ( c ) are constants, and ( a <br/>\ne 0 ). The solutions (or roots) of the equation give the values of ( x ) that satisfy the condition.", "---", "### Step 1: Identify Coefficients", "For the equation ( x^2 - 5x + 6 = 0 ), we identify:", "- ( a = 1 )\n- ( b = -5 )\n- ( c = 6 )", "---", "### Step 2: Factor the Quadratic Expression", "We seek two numbers that multiply to ( a \ imes c = 1 \ imes 6 = 6 ) and add up to ( b = -5 ).", "The numbers ( -2 ) and ( -3 ) satisfy these conditions because:", "[\n(-2) \ imes (-3) = 6 \quad \ ext{and} \quad (-2) + (-3) = -5\n]", "So, we can factor the quadratic as:", "[\nx^2 - 5x + 6 = (x - 2)(x - 3) = 0\n]", "---", "### Step 3: Apply the Zero Product Property", "If the product of two factors equals zero, then at least one of the factors must be zero. Therefore, set each factor equal to zero:", "[\nx - 2 = 0 \quad \Rightarrow \quad x = 2\n]", "[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "---", "### Step 4: State the Roots", "The roots of the equation ( x^2 - 5x + 6 = 0 ) are:", "[\n\boxed{x = 2} \quad \ ext{and} \quad \boxed{x = 3}\n]", "---", "### Why Factoring Helps", "Factoring simplifies solving quadratics without needing complex formulas. It is particularly useful when the quadratic expression factors neatly into binomials, as in this case. For equations that do not factor easily, the quadratic formula serves as a reliable alternative:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "---", "### Conclusion", "Solving the quadratic equation ( x^2 - 5x + 6 = 0 ) yields the clean, rational roots ( x = 2 ) and ( x = 3 ). Understanding how to factor and apply the zero product property enables quick and accurate solutions—an essential tool in algebra and beyond.", "Whether you're a student, teacher, or self-learner, mastering quadratic equations like ( x^2 - 5x + 6 = 0 ) strengthens your problem-solving foundation. Keep practicing, and mastering these roots will become second nature!"]
