A quadratic equation \( x^2 - 5x + 6 = 0 \) is given. What are its roots?
["Understanding the Roots of the Quadratic Equation ( x^2 - 5x + 6 = 0 )", "When solving quadratic equations, identifying the roots is essential for understanding the equation’s behavior and applications in math, engineering, and science. One of the most commonly studied examples is the quadratic equation:", "[\nx^2 - 5x + 6 = 0\n]", "But how do we find these roots? And what are they exactly? Let’s explore step by step.", "---", "### What Are the Roots of a Quadratic Equation?", "The roots of a quadratic equation ( ax^2 + bx + c = 0 ) are the values of ( x ) that satisfy the equation — that is, the points where the parabola intersects the x-axis. These roots can be real or complex depending on the discriminant ( b^2 - 4ac ). When real and distinct, they represent two unique solutions; when equal, one repeated root occurs.", "---", "### How to Solve ( x^2 - 5x + 6 = 0 )", "There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Given the simplicity of this equation, factoring is the quickest and most straightforward approach.", "We want to express ( x^2 - 5x + 6 ) as a product of two binomials:\n[\nx^2 - 5x + 6 = (x - a)(x - b)\n]\nWe look for two numbers ( a ) and ( b ) such that:\n- ( a + b = 5 ) (coefficient of ( x ))\n- ( ab = 6 ) (constant term)", "Checking integer pairs that multiply to 6:\n- ( 2 \ imes 3 = 6 ) — and ( 2 + 3 = 5 ), which matches perfectly.", "Thus, we factor the equation as:\n[\n(x - 2)(x - 3) = 0\n]", "---", "### Finding the Roots", "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]", "---", "### Conclusion: The Roots Are 2 and 3", "Therefore, the roots of the quadratic equation ( x^2 - 5x + 6 = 0 ) are:\n[\n\boxed{2 \quad \ ext{and} \quad 3}\n]", "These solutions indicate that the equation = 0 when ( x = 2 ) or ( x = 3 ). Understanding these roots is crucial not only for solving equations but also for interpreting real-world problems modeled by quadratics — such as projectile motion, optimization, and financial analysis.", "If you're learning algebra, mastering quadratic equations and their roots lays a strong foundation for more advanced math topics. Use online quadratic formula calculators or practice with factoring exercises to strengthen your skills.", "Keywords: quadratic equation roots, solve (x^2 - 5x + 6 = 0), factors of quadratic, discriminant, Vieta’s formulas, algebra practice, quadratic formula, real roots, math solutions, educational math."]
