A quadratic equation \(x^2 - 5x + 6 = 0\) needs to be solved. Find the roots of the equation.
["# Solving the Quadratic Equation (x^2 - 5x + 6 = 0): Find the Roots", "When learning algebra, solving quadratic equations is a fundamental skill. Among several standard forms, the equation (x^2 - 5x + 6 = 0) is a classic example that helps students understand factoring, the quadratic formula, and root identification. In this article, we will explore how to solve this equation step-by-step and find its roots clearly and accurately.", "## What is a Quadratic Equation?", "A quadratic equation is any equation of the form (ax^2 + bx + c = 0), where (a), (b), and (c) are constants and (a <br/>\neq 0). The general solution involves finding the values of (x) that satisfy the equation — known as the roots or solutions.", "## Given Equation", "We begin with:\n[\nx^2 - 5x + 6 = 0\n]", "Here, (a = 1), (b = -5), and (c = 6).", "## Method 1: Factoring – The Simplest Approach", "Since the equation has integer coefficients and a small constant term, factoring is an efficient method.", "We look for two numbers that:", "- Multiply to (c = 6)\n- Add up to (b = -5)", "Let’s list factor pairs of 6:", "1. (1 \ imes 6) → sum = 7\n2. (2 \ imes 3) → sum = 5 → since both are positive and we need a negative sum, try –2 and –3 → sum = –5 ✅", "So, the equation factors as:\n[\n(x - 2)(x - 3) = 0\n]", "## Step-by-step Solution Using Factored Form", "Set each factor equal to zero:", "[\nx - 2 = 0 \quad \Rightarrow \quad x = 2\n]\n[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "## Roots of the Equation", "The roots of (x^2 - 5x + 6 = 0) are:\n[\n\boxed{x = 2} \quad \ ext{and} \quad \boxed{x = 3}\n]", "## Why Factoring Works Here", "Factoring is especially effective when the quadratic is easily decomposable with integer solutions. This equation fits perfectly, allowing quick identification of roots without complex calculations.", "## Alternative: Using the Quadratic Formula", "For completeness, here’s how we could use the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plugging in (a = 1), (b = -5), (c = 6):\n[\nx = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)} = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm \sqrt{1}}{2}\n]\n[\nx = \frac{5 \pm 1}{2}\n]\n[\nx = \frac{6}{2} = 3 \quad \ ext{or} \quad x = \frac{4}{2} = 2\n]", "Same result — confirming our factored solution.", "## Conclusion", "Solving (x^2 - 5x + 6 = 0) is straightforward by factoring, yielding roots (x = 2) and (x = 3). This classic quadratic teaches essential algebraic techniques and serves as a foundation for more advanced equation solving. Whether you use factoring or the quadratic formula, mastering such equations strengthens core math proficiency.", "---", "Keywords: quadratic equation, solve (x^2 - 5x + 6 = 0), roots, factoring method, quadratic formula, algebra solutions, algebraic equation.\nMeta Description: Learn how to solve the quadratic equation (x^2 - 5x + 6 = 0) by factoring, discover roots, and understand key algebraic methods—perfect for students and math learners."]
